If objects in motion experience time differently, how does my body stay synced when I move my legs or arms?

The relative speeds of the parts of the body are much slower than the speed of light.

If you wish to calculate the ratio between the times of different clocks, you can calculate it. The ratio is: $sqrt{1-frac{v^2}{c^2}}$, where $v$ is the relative velocity, and $c$ is the speed of light. If you try plugging in the velocity of your hands, you'll find that the factor is very close to $1$, because the speed of light is much greater than the velocity of your hand. So the different clocks tick pretty-much the same.

The problem comes when something is moving very fast, like satellites, which do have to account for the effect (like GPS satellites).

The relativistic affects of time dilation become noticeable when speeds become high. Especially close to the speed of light. That means when we are talking about the relatively slow speeds in moving our body parts, relativistic effects become so negligibly small, that they can be ignored.

I think the best way to see the answer to this is to calculate the order of magnitude of the time dilation effect your legs experience when they move relative to your body.

Let's say for simplicity's sake that they move at 1 $mathrm{m , s^{-1}}$. The magnitude of relativistic effects like time dilation is measured by the Lorentz parameter, defined by

$$gamma = frac{1}{sqrt{1 - frac{v^2}{c^2}}},$$

where $v$ is the object's speed and $c$ is the speed of light, $3 times 10^8 mathrm{m,s^{-1}}$. An observer sees a clock on their moving leg with period $gamma Delta t$, if the clock's period is $Delta t$ in its rest frame.

If you plug the numbers into the calculation above, you find that the effect of time dilation for one's arms or legs is of order $10^{-15} %$. This is definitely too small to have an observable effect.

how does anything ever stay synced ?

Not sure what you mean by "stay synced". Different parts of your body maintain their structural integrity at the atomic level because of the electromagnetic forces between atoms and molecules. This simply involves the exchange of photons (the force carrier for the electromagnetic force) over very short distances - no "syncing" is required. Similarly, nerve impulses to and from different parts of your body are chemical signals sent down nerves, which also ultimately depends on the exchange of photons at an atomic level. Again, no "syncing" required.

In computer science terms, the body is an asynchronous system. There is no master clock in the body that says "hey, arm, you're a femto second behind everyone else".

Different parts of your body do pass through time slower than others.

This is actually how gravity works. Time runs slower the deeper you are in a gravity well. Think of it as like floating on a river where the current near the banks is slower than the current in the middle of the river. The different speeds of different bits of the boat cause it to twist in the direction towards the slower flow. Similarly, matter in a gravity well is pushed towards the areas of slower time. This difference in the flow of time is what we call space-time curvature. The path you follow in freefall is the path that maximises the time you experience, given these distortions in its rate of flow.

An accelerated frame of reference (like rotating your body) looks like a gravitational field, by the equivalence principle. The different time dilation of different bits of your body distort the flow of time in the same way a gravitational field does. So you feel a gravity-like 'centrifugal force' outwards towards the region of slower time. The inertial forces you feel as you swing your arms around are (from a certain point of view) caused by the forces imposed by the differing flow of time.

To explain more precisely, we need to use quantum mechanics and think of matter as a wave that progresses in the direction of motion through time. When you are stationary, all parts of the wave beat in sync. When you are moving, the wavefront is tilted in time and the waves for different bits of the body are phase-shifted relative to one another. When time slows down in a gravitational field, it acts on the wave like a refractive medium, like light through glass, slowing the wave down, that causes the wavefront to bend and change direction. And that changes the velocity. Although the distortion of time is super-tiny, because the frequency at which the wave is oscillating is so super-fast, it only takes a tiny distortion to shift different parts of the wavefront by many wavelengths.

There is simply no need to stay 'in sync' - for what?

If you swing your arms, their time goes a (microscopic) bit slower, so your body is simply a bit older than your arms.

For any communication between your body and your arms (where 'in sync' might be important), the speed you are able to swing your arms results in such small amounts of time difference that it makes no difference - if your hand hits an obstacle, it doesn't matter if it takes 0.08 seconds or 0.08000000000000001 seconds for your brain to feel pain.

Your arm is traveling at a tiny percentage of the speed of light. The speed of light 300 million meters per second, so even if your arm were moving at a ridiculous speed of 300 m/s, it would still be only one millionth of the speed of light. So even if there were a linear effect, it would be tiny. But the effect isn't linear. The exact amount is given by the Lorentz factor, but for small speeds, it can be approximated as $frac {v^2}2$. This shows up in the formula for kinetic energy: in Newtonian physics, it's $frac {v^2}2m$, which is an approximation of the relativistic amount. Using this approximation, your arm would be off by one part in 2 (million)^2, or one part in 2 trillion. Over the course of sixty years, your arm would experience one fewer millisecond. If you only reach 3 m/s, it would be one part in 60 quadrillion.

If I move my arm really fast ...

We humans think we can move quickly. We can't. When compared with the micro-machines that make up our bodies, we are actually huge giants moving incredibly slowly on a low-gravity planet. We only think we move fast because our brains are slow. Luckily they are fast enough to keep our bodies balanced.

Of course "slow" is comparative. But, compared to the finest divisions of time, we live for an eternity.

physicists have successfully recorded an internal atomic event with an accuracy of a zeptosecond (a trillionth of a billionth of a second). Their measurement is the smallest division of time ever to be observed and recorded by humans.

https://futurism.com/physicists-have-measured-the-smallest-division-of-time-ever-observed

The relativistic effects of moving our body are negligible (almost zero) compared to the speed of light (the fastest thing possible). We are kept in balance by nerve signals.

Approximate speeds in metres per second

Light 299 792 458 metres per second

Nerve impulses 120 metres per second (from brain to muscles)

Humans (Usain Bolt) 12 metres per second (approximately)

Answer

The brain has constant feedback from the inner ear for balance, and proprioception tells us where our arms, legs, and other parts are at any given instant. Because the brain works faster than our muscles, it can keep recalculating all the time to prevent us falling over. In fact the brain has numerous ways of synchronising movement. I could write an essay on this, however maybe those questions would be better answered under Biology.

Forgive me for answering a question with a question, but what makes you think that your limbs do stay "in sync" as you put it?

Let's examine one particular molecule in your wrist. As you swing your arm, the atom of this molecule that's closer to your arm probably experiences one or two more Planck Times, and thus has a tiny fraction of a femtogram less mass, than the atom that's closer to the hand. Molecules experience trillions of times more force than that just existing in a world with other molecules to bounce off of.

The fact that one atom of this molecule is now "younger" than another doesn't matter in the slightest. In fact, the concept of "age" has no meaning for an atom at all. It will continue to participate in every chemical reaction it would have otherwise, exactly the same.

Let me start by saying that time is running objectively slower if they are moving in a gravity field (be it artificial or real). Two objects in relative motion with constant velocity don't experience an absolute time difference. Only a relative one. The objects are symmetrically treated. In the twin paradox it is examined what happens if one of the two objects in constant relative motion suddenly is changing its motion to reach the other object. Depending on how they got their relative motion in the first place, the absolute times on both objects will differ.

Well, the question. Imagine the speed of light to be 1 m/s. And let's assume your entire body finds itself in a place where it resides in a somewhat static state. No external factors affect your life.

Let's look at your blood flow (or your whole body). Obviously, when you move one arm, it gets accelerated, so there is an artificial gravity operating on your arm. This means the time in your arm is not in sync with the time in the rest of your body (which means the time in your body and in your arm are running at a different pace). When your arm is moving at constant speed again, both times are in sync again, but the time in your arm runs behind the time in your body.
What does this mean for your bloodstream? During the acceleration of your arm, the time in your arm is running objectively slower. This means that less blood is streaming through your arm wrt to the blood running through your body (let's assume your body to be at rest; there is some resemblance here with the twin paradox).

What will happen with your blood? Actually, nothing. The blood may be moving slower through your arm, but the amount stays the same (it's somewhat similar to missing an arm). If you also start moving your other arm, your legs, and your head (all to and fro, though this will be very tough as the speed limit is 1 m/s and the mass of your arms, legs, and head will get enormous so you are bounded in your movements) again, nothing will happen. Although the speed of your blood is different due to the "to and fro" movements of your arms, legs, and head, the amount of blood stays the same in all parts of your body.

What does change is the aging of your arms, legs, and head (when you keep moving them in an accelerated way; the accelerated movements of your terms, legs, and head experience artificial gravity which is equivalent to real gravity and as you probably know, time slows down in a gravity field).

So they won't age (or at least much less than your torso) while your torso is aging relatively very fast. So your torso gets old while your arms, legs, and head will remain almost constant in age. Your torso gets wrinkled, old (your vital organs start to disfunction, etc.), while your arms, legs, and head remain young (your brain can send signals to your arms and legs through your fast-aging torso). Very disturbing! And according to your very slowly aging brain, this happens in a flash. What a trip! Phfffuuu...
So, luckily, the speed of light is not 1 m/s, but about 300 000 000 m/s!!!

We know that orbiting satellites experience time dilation significant enough to be measured; we have even observed it in aircraft carrying atomic clocks and flying round the world in opposite directions; the Earth's rotation carries the atmosphere with it, so one plane moves faster than the other with respect to the inertial reference frame and the two clocks register different times for a journey of the same length. This is clearly what the OP means by "losing sync".

A better example to visualise this question might be a space elevator. Here, clocks at the base and apex of the elevator run at different speeds, they do indeed lose sync. Yet they are physically connected by the structure of the elevator. It is just the whirling-arms scenario writ large enough for us to measure today. So, how can this loss of sync happen?

The answer is that time is not absolute, it relative to the observer. This applies to both the passage of time and to given points in time. An observer beside each clock will see the other clock running at the wrong speed and indicating the wrong time. But that is just their viewpoint; the other observer sees it the other way round.

Thus time synchronisation is not a general rule, it can happen only when two objects lie in the same inertial reference frame. Spin round at any speed and you create your own inertial frame distinct from that of your centre of rotation.

When you spin round, your hands will theoretically age fractionally faster or slower according to how they are moving with respect to the Earth's rotation, but the effect will be ngeligible. If you stand on the equator the variations will cancel exactly.

There will be an even smaller effect arising from your hands' velocity with respect to the mass of your body, but it is probably unmeasurably small.