I think it is important to remind people that booking a ticket as one journey is important when transiting somewhere. If you buy the tickets separately and your first flight is delayed, you do not receive help re-booking or if you need a hotel etc. So if I were travelling from Newcastle to Budapest and there was no low cost direct flight, if cost is what is important to you, I would not use any airlines that do not allow you to purchase your tickets as one journey.
Looking at the original question, I think ease of travel was what this passenger was after and as already stated I think the BA option would be the easier and more enjoyable routing.