Bit of a dilema and want to ask you folks for some advice

Am on a J "non changeable" + non refundable albeit pretty expensive flight from Italy - first flight out Thursday 1 april ITALY -> LHR

This is the return sector (outward bound already taken)

Due to the strike am unable to get a connection (off to States) until the Friday evening

Ironically the flight I am on according to seatcounter is TOTALLY full - what are my possibilities of trying to get BA to move my flight to the Friday so as not to stay 36 hours in London? Is it worth a go?

Tried the Euro line earlier but couldn't get through and would hate to wait an hour to hear - "sorry - were you on the 31st... But..."

Are there any other phone lines (gold etc) that I could use?

Has anybody else tried to move flights first day "outside" BA bracket? Are they flexible?