I’m assuming this is an ex-EU and on two tickets? LHR-BUD and BUD-DOH-HND?

If it is a straight forward Club Europe ticket then it should be eligible for free Same Day Change (SDC). So as long as there are CE seats on an earlier flight you should be able to switch to it from midnight on the day of departure.

The exception to this are some corporate and other special tickets, tickets booked as a package including through BA Holidays and if you have taken advantage of a promotional upgrade.

The SDC may be able to be done online, you may need to call or it may be possible to do it at the desk at BUD. Others may have experience on whether BUD agents can process these.

Edit: somehow transposed Budapest and Bucharest in my mind. Corrected, but points still apply