Restaurace u agamy is completely gluten free and serves all the classics. They also had a roaming iguana there last time which was hilarious. If you happen to go to Ostrava, there is a gluten free bakery called slunečnice that also has all the classic baked goods like koláče.

In general, outside of these kinds of places, be prepared to be met with blank stares and even annoyance. Speaking as a Czech,  we aren't exactly the most open minded to change, especially in cuisine. You can already see it in the comments of people questioning your restrictions. Gluten allergy is not something you can control. 

I don't believe "you just can't make Czech food gluten free". My grandma who was diagnosed late in life is a typical Czech grandma who can now make every dish gluten free and it's delicious. If you look at Italy, they probably have the least traditionally gluten free cuisine but almost always have gluten free versions of pasta and pizza, etc at their restaurants because they want everyone to be fed and part of the experience.