Very unlikely. Most places in 'eastern' Europe still require you to actually sign and pay for the ticketup front.

However, with the RTW rules, you are permitted to buy the ticket up to one hour prior to departure!

So, get BA in UK to make the entire reservation, and then say that a 'relative' will pay for it in Budapest at the airport, so BA will price it as ex-BUD and not ex-LHR. Then take a flight to BUD, and go to the BA ticketing counter at the airport. Issue the ticket and check in for your return flight.