Welcome to FT and the BA Board.
Missing the outbound flight will automatically cancel the whole ticket. This is fairly standard airline practice and not BA specific.
You should be able to reclaim the APD and US Transportation tax as well as airport fees.
For a Y fare this is around £160 less any BA admin fee.
You will have lost everything else - the base fare and the BA surcharges (which is not a tax)
Why on earth didn't you do on line check in and leave it so close to departure to check-in and arrive at LHR?