Not sure how efficient BA is, but many airlines will move found items to contract 'lost property' companies. (I think they use bagport and LostProperty.com)

The good news is that wallets are of little use to others AND people tend to empathize with the situation- so they get turned in and returned (if found). The bad news is that some of these lost item companies can charge $$$ to return things. For a wallet or iPad, thats fine with most. For a kids old rag dolly, 60GBP might be a bit much.

Id fill out the online forms. Costs nothing.

I left a wallet on an AA seat once headed into PHX late. Arrived at DEN rental car sans wallet. Eventually got to a hotel; filled out the form. In the AM, there was an email saying they found it. Had the wallet back in less than 48hrs, just in time for a departure to EU....