You definitely cannot do this - if I remember correctly, this is not permissible and I believe it can end up in the hands of the Lost Property service. That is something you would really rather avoid (they'll charge you a retrieval fee for a start - LHR lost property is not operated by BA and the system is not reported to be very efficient, potentially resulting in your parting company with you bag for longer than you meant to).

There is nowhere to store your bag airside.

There is Excess Baggage Company that provides storage facility but it is landside.