Sory to hear about your problems. Try to get through to BA and ask for rerouting on LH /LH Group flights if no seats with BA available. BA will not rebook to any other carrier.
Another alternative is to take the train to Lyon (it is a short ride and there is a TGV station right at the airport) or even to Paris to get on a BA flight from there.

EDIT: if no acceptable flights, take the train to Paris and Eurostar from there to London. Keep tickets/receipts for reimbursement from BA afterwards. If time allows it, ask for seats on Sunday. Hotel in Marseilles or Lyon to be reimbursed by BA.