Your lounge access entitlements are clearly printed on your online Boarding Pass, and also clearly set out during the purchase and Manage My Booking process.
Had you any checked luggage you should also have been told by the agent where the lounge was, even if you didn’t use the F check in (and if travelling internationally you would have had to have your Passport checked, so should have encountered a BA employee at some stage prior to security?)
Every BA First Class passenger is entitled to enter the Concorde Room, and has access to all the five lounges in T5.
It’s unfortunate you did not see the right information, but when in doubt it’s best to ask someone to confirm your understanding, rather than leaving feeling you “deserved better”.
The baggage retrieval processes at T5 are among the world’s most advanced, and last time I arrived with luggage my cases were on the belt before I arrived in the Baggage Hall (about 10 mins from disembarking).