What the AA person told you is plain wrong. Since there is C space available what you will do is go ahead and buy your ticket in economy (just make sure it is not basic economy) and once the flight is ticketed which can take 1-2 hours you will phone in AA and tell them you will like to upgrade with miles plus copay.
The agent will put in your request and if C space is available they will confirm your upgrade on the spot. No need to wait until 24 hours. You will have to pay additional taxes as UK has a luxury tax for premium cabins. This will be an additional ~$85 if I remember correctly.
I would avoid booking awards on BA even if there is saver inventory. Fuel surcharges are ridiculous on BA and will probably run you $500-600.