Hi there,
I'm Hungarian, do not live in Budapest, in fact have never lived there, but have a flat there, so visit the city every now and again.
I have looked up international school on the internet, as my children are in the UK and toying with the idea of moving home. There are plenty in the capital that follow the UK curriculum. And there are other international schools too, with, presumably different curriculum. They are, of course, all fee paying. I cannot comment on how good they are, no first hand experience of them.
There are alternative schools too, eg. Waldorf.
In Budapest you will have no problems, youngsters speak English, everything can be found in English.
The only big difference, if you are more adventurous, and want to send your children to local schools, that in Hungary the primary education starts when the child is over six years old (just like in most European countries), so two years later than in the UK. Kindergarten is for three years, between 3 and 6.
Hop it helped a little.