Paddington should have added you to the standby upgrade list. If they did not (and it sounds like they did not), then I would call BMI's telephone check in line on 0870 600 6868 , explain the situation and ask to be added to the standby upgrade list.

When you get to the lounge, check the situation with the staff there - if the upgrade has cleared they may well give you your new boarding pass there.

Eligibility for the availability of seats depends on your status (Gold or Silver) but also the time you were added to the upgrade list, which is why it is important to phone them up.

This answer assumed you are travelling on a BD flight - if not then the procedure will differ, but you should phone the operating carrier's LHR office and get yourself on the list.