If you presented yourself for check in at 55 minutes before scheduled flight departure, then BA should not have turned you away. As for telling you the PHL flight was full, I suspect it was because it was closed for check in, so no seats were for sale. I've had the same when trying to book on to a flight which I subsequently found to be cancelled -the agent said the flight was "full" -I think it just shows up zeroed out on their screen, so I doubt the agent was trying to mislead you.

Yes, it's definitely worth contacting BA. 55 mins, whilst not ideal by any stretch, is within the required check in time, so you should have been accepted for travel. You were effectively denied boarding, and then delayed by four hours. I would hope you would get an amicable resolution from BA.

I would say for future reference -if in doubt, shout! I don't mean raise your voice -I mean ask for a supervisor. If the supervisor isn't giving you the answer you want, ask for the duty manager. That's why these people are there, and the Supervisors and DM's at BA LHR are in my experience, without fail, always proactive and helpful -I've always felt that they've done their best to help. Sometimes they just cannot get you what you want. Even so, they will always try to help, as long as you are not rude, which I'm sure you're not

Doesn't help you now though -so yes, write to BA as factually as possible. Also try to produce something like a car park receipt/ticket showing what time you arrived at LHR, and if pos, if you did use the SSCI machines, ask them to check the log of when you actually checked in. Although if the flight was oversold, you'd have been referred to a check in assistance desk by the machine.