There is only one direct flight a day between the UK and Budapest with Estonian Air:
- Departing Gatwick at 1800hrs to arrive Budapest at 2255hrs
- Departing Budapest at 1620hrs to arrive Gatwick at 1715hrs
(and no direct flights on a Saturday)
Because there's a lack of direct flights, try and book as far in advance as possible.

Estonian Air - http://www.estonian-air.ee/?keel=eng

Should flights be unavailable to Budapest, book your flight via Helsinki instead with onward connection via flight or high-speed ferry (ferry is only 1hr 30mins).