Maybe I'm missing something... but is it or is it not correct that BA carried the OP to Hungary with no valid travel documents in his possession?
And now the absence of such documents - very clearly an oversight - are preventing the OP's return to his adopted home of the UK.
It seems almost that it would have been better if the OP had realized during the flight to Hungary that he had no documents, and then had his admission to the country refused, so that at least he could go back home to get his passport or whatever other documents might have been left behind in the UK.