I think that we pretty much all agree that it would have been more prudent for the OP to take a private test and, had the OP come here to advice prior to the trip, it is likely that there would have been near universal advice to take a private test. But this is not where we are and all the "coulda", "shoulda" in the world are irrelevant at this stage. At this point, the question is whether he can salvage the holiday and/or whether BA could lawfully deny boarding or not, not whether it would have been more prudent before the flight to take this or that course of action.