Of the three major alliances, you've covered OneWorld with BA, and Star Alliance with LH/SN. There's also Skyteam with KL or AF from Newcastle via AMS or CDG, though sometimes they can be a bit pricey.
Loco options include Wizzair, which goes to Budapest thrice daily, but from Luton; and easyJet from Luton or Gatwick. Obviously, flybe can take you from NCL to LGW, but Luton is a bit of a hike.
Don't forget that Jet2.com also flies to Budapest from Edinburgh and Manchester. Quite good fares still for July, too.