Body got skinny and gaunt; caused by not eating meat any more?

In order to address this question, you need to supply more information.

You mentioned what you ate, but you didn't quantify it. First do that: How many calories do you eat per day?

You mentioned that you cycle, but you didn't quantify it. Second do that: How many calories do you burn per day?

If you're burning a lot of calories (easy to do on the bicycle), then you probably need to consume much more than the average recommendation of 1,500 - 2,000 calories per day.

When I was in my 20s, I would cycle a lot, and I often had to eat 5,000 - 7,000 calories per day. I ate mostly pasta. So much pasta. I did not get gaunt. My leg muscles ballooned in size, and I was in absolutely the best shape of my life.

But until you or someone sits down and does the math, nobody can help you.

You need more calories

If you are loosing weight, and you don't want to be, you aren't eating enough calories. You can use the Dietary Guidelines for Americans (they work for Germans too) to estimate how many calories you need to eat in a day to maintain your weight.

You should start tracking how many calories you're eating, and steadily increase the amount until you stop loosing weight. You can track your calories with pen and paper, but there are also applications to help you track the calories and other nutrients you eat, such as Cronometer. Most food you buy will have nutrition information printed on its package. For food that doesn't come with nutrition information, such a fresh produce, you can look the information up online. Once you figure out how many calories you need to maintain your weight, you can start eating more to begin gaining back weight.

How do I eat more calories?

Below are some strategies I've successfully used for gaining weight despite my low appetite:

• Add peanut butter to things.
• Add walnuts to oatmeal.
• Snack on unsalted nuts.
• Have a big breakfast. I went with 8oz of scrambled tofu plus 1 cup of oatmeal with raisins and walnuts.
• Eat four meals per day.
• Eat big portions. Have a specific calorie goal in mind per meal.

What about protein?

Other answers mention other nutrients, in particular protein. These nutrients are important, and it's important that you get enough of each micro and macro nutrient, but nutrient deficiencies do not typically cause significant weight loss. Weight loss and gain is driven predominantly by caloric intake.

For protein specifically, it's generally easy to get enough if you are eating enough calories, and not many of those calories are from very low protein "junk foods", such as potato chips, sugary sodas, etc.

Addendum

Note that while eating too many or too few calories is the most common cause of weight gain or weight loss, there can be other causes, such as diseases like Coeliac Disease. If you keep loosing weight even after significantly increasing your calorie intake, you need to see a doctor.

You lack protein and probably also B vitamins, your diet is not balanced. You miss the plant proteins, which are found in beans, lentils, chickpeas, tofu...

It is very important to have a balanced diet. I suggest you go to a good nutritionist, not to a doctor.

I hope you're not seeking nutritional/medical advice.

my arms and legs are noticeably thinner

Ok? Do you want thick limbs? There's a plethora of vegan bodybuilding cookbooks out there. Additionally, Fill Up! A Practical Healthy Vegan Weight Gain Guide by Naluwagga has poor grammar, but is a decent guide.

I've noticed that this year I'm not as fit as the years before: in particular after a 40-km bike ride

High carbs are key to cycling. Perhaps look into an auto/biography of vegan cyclists like (1) Dotsie Bausch, (2) Molly Cameron, (3) Adam Hansen, (4) Czesław Lang, (5) Christine Vardaros, etc.

Is my current diet unhealthy?

You haven't said enough to actually know. Besides, this isn't the place to ask such a question. Instead, work with a nutritionist that specializes in vegan or sports nutrition.

I've thought about drinking protein shakes - is that a good idea?

Perhaps log everything you eat for at least three days on Cronometer. And I mean everything. You will likely need measuring cups. This is a visual way to see and track how much protein you're getting--and just about any other nutrient. Example:

And how quickly should I expect to see any results?

Your results in cycling performance? I wouldn't rely on first blaming diet for athletic performance. (I'm not a certified nutritionist.) Other primary performance inhibitors are poor sleep hygiene (e.g., consistency and punctuality) and stress.

1. Track cycling, Dotsie Bausch went vegan three years before the 2012 Olympic Games, where she medaled, then went on to achieve eight US national championships, two Pan American gold medals and a world record.
2. Cyclo-cross cycling - Molly Cameron.
3. Road cycling - Adam Hansen.
4. Road cycling - Czesław Lang.
5. Road and cyclo-cross cycling - Christine Vardaros.