I work for London Transport,, and take it from me, don't hand it in to them, take it to the Police Station.
A couple of rarely known things.
First, they will make no effort to trace the owner.
Second,
If the owner contacts them, they charge a percentage of any moneys in the wallet, before the owner can have it back.
Third, if you take it to the Police Staion, you've done the honest thing by handing it in, but after six months, if its not claimed, its yours.
The money, not the oyster card.