Those with a proper aviation knowledge in here can correct me, but I thought it was a combination of Concorde-served flights becoming increasingly unprofitable unless you racked up tickets to ludicrous levels, and Airbus growing disinterested in offering a maintenance/ parts replacement service for BA and Air France.
The thing is, whereas BA and Air France keeping a regular 6-aircraft daily schedule might have indeed become too difficult to maintain, given that there is no shortage of 1%ers charting private business jets for considerably more money, they probably could have kept at least one or two birds flying the London/Paris to NY route at 15k per head, which surely would have been profitable. But I guess the overall revenue brought to the company’s accounts would have been a footnote.