I travel more than most.

UK is definitely fine, whatever they tell you! Vienna is definitely fine (and I think you may have been misinformed about leaving the airport). Budapest is definitely fine, but unlike VIE and LHR there is some control of transfers within BUD airside. One of the problems with contacting airlines or airports - and this is a very general point - is that the people answering the questions don't tend to do much travelling themselves, hence FT is a better source of information, since it's real world.