Hi, this is probably to late but here it goes anyway.
I've been in budapest and, like you said, you see many bikes around. But most of them ain't for MTB'ing.
I wasn't there to do MTB but there are places in the city where you may find good trails to do some MTB'ing. You have some montains on the buda side, not sure if it is buda or pest, they are easy to spot. There a big statue on the top of one of them, thats the statue of liberty. There you can ride your bike with some cool trails.
I've been in budapest and, like you said, you see many bikes around. But most of them ain't for MTB'ing.
I wasn't there to do MTB but there are places in the city where you may find good trails to do some MTB'ing. You have some montains on the buda side, not sure if it is buda or pest, they are easy to spot. There a big statue on the top of one of them, thats the statue of liberty. There you can ride your bike with some cool trails.