Here in Brazil we have "borracharias" a repair shop with many tires where they replace or fix flat tires. "Borracha" means rubber. Neither Google Translate nor Wikipedia could help me to find an English translation for "borracharia".
In the USA, Japan, and Taiwan, four types of shops specialize in fixing flat tires:
Gasoline stations that also have shops that repair cars or motorcycles
Auto repair shops
Motorcycle repair shops
In Japan, though, you usually can't get a flat tire on a motorcycle repaired at an automobile repair shop, and in Taiwan, there are so many motorcycle repair shops that you wouldn't want to take your bike to an auto repair shop to have the tire fixed.
I can't think of a single English word used to describe these places.
There is no single English name for a repair shop specializing in fixing flat tires, as Bill points out.
Names for business types, like names for food. are intensely cultural and vary widely from locality to locality, and thus from language to language.
In English one only finds names for businesses that are commonly encountered in Anglophone areas. This is just as local as anywhere else. In Mexico, for instance, to refinish a desk, one buys varnish at a tlapateria but new drawer pulls at a ferreteria; in the U.S. one buys both at a hardware store.
There are stores in Mexico that just don't exist in the U.S, like a refaccionaria, where one buys spare parts (refacciones -- in Mexico; in Guatemala refacciones means 'snacks'). They stock spare parts for just about anything, from cars to kitchen blenders. There's nothing like that in the U.S.
In the UK the general term is garage, which can mean 3 things:
A building next to your house where you keep your car;
What Americans call a gas station, ie a place selling fuel;
Any place where mechanics work repairing and servicing cars.
In the UK tyre repairs are dominated by a few well known national companies so you'd be more likely to say 'I took my car to - [name of company]. Some of these deal only with tyres, others do other repairs. If you're talking of a small, privately owned repair shop you would probably call it a garage.
And yes, we do spell 'tyre' with a 'y'. 'Tire' is, you know, tired. Zzzzzzz.