Ryanair doesn't fly to Budapest from any London airports. You'd have to fly to Dublin, then fly the next day (the 24th) to Budapest. Under the current sales, that would cost 9.99 pounds plus 38 euros ($71 plus all Ryanair fees).
EasyJet has a flight leaving in the afternoon of the 23rd (4:45 PM) for 38.99 pounds ($63) (plus 8 pounds if you check a bag up to 20kg).
Malev Hungarian also has an evening flight, leaving at 6:50 PM, for 23,400 florents. It's a bit more expensive, about $107.
I'd say easyJet is your best option. You're clearly not going to get low fares flying same day or next day if you book it when you land in London so I'd book now.