Object with a uniform circular motion

The object is not rotating around its axis.

Even though the instant center of rotation lies outside the object, that doesn't mean that it's not "rotating about its own axis". The rotational period is identical when measured from any inertial (non-rotating) reference frame.

In the same way, we say the earth rotates on its axis even though it has other complex motions (depending on the reference frame).

In all these non-rotating frames, we will see the rotational period of the object remains the same before and after the string is cut.

If we suppose an spherical object and focus in an volume element close to the string, its distance to the rotation center is $R-r$, where $R$ is the distance from the rotation center to the center of mass, and $r$ is the distance from the element to the center of mass.

For an element at the opposite side of sphere, its distance is $R+r$.

When the string is cut, each element has a different momentum: $p_i = dmomega (R-r)$ and $p_o = dmomega (R+r)$. And the momentum of an element in the CM is $p_{cm} = dmomega R$.

For the object follows a trajectory without rotating, both momentum at the periphery should be changed, to match the momentum of CM, and that would require a force.

As there is not such a force, the object keeps rotating as it was before.

When the string is cut there is now no external force acting on the object therefore the object will continue in a straight line that is tangent to the circle at the point where the force stopped acting.

Blue5000 and BowlOfRed are both correct. Blue notes the conservation of linear momentum, or Newton's 1st Law. It look's like you're good with that based on your conversation with Red. Red focuses on a topic that's a little less intuitive (at least for me). I want to expound on their answer with a familiar example: The Moon's orbit around Earth. The same side of the Moon always faces Earth. As a result, it's tempting to say that the Moon is not rotating. However, the Moon is rotating with an period equal to its orbital period around Earth. This is a consequence of tidal locking, and it is the equivalent of the string attached to the object in your scenario. That should mean that the object in your scenario is rotating about its axis with an angular speed equal to that of its CM around the center of the string.

My guess is that, when the string is cut, the object will travel with a constant linear velocity (as noted) and continue to rotate around its CM with that same angular speed.

Illustration to help visualize the Moon's rotation.

Image comes from http://ipfactly.com/you-always-see-the-same-side-of-the-moon/

Wikipedia's article on Tidal locking.