Generally both BA and Malév airbridge at BUD, it's the smaller planes that they seem to coach there.

Customs (if you even notice it) and passports are quick especially if CE. Hungary is in Schengen so there isn't a high proportion of travellers that need to be checked. Can't comment on baggage handling 'cos I don't do that, but I count on 15 minutes to get landside in terms of public transport schedules, and I usually beat that.