I would check out on schedule and head to the airport. There's a good selection of restaurants etc. landside, and the BA checkin will open at about 2.30 for the early flight. You can go through and wait in the lounge, which is comfortable enough and offers limited hot food.

If you decide to wait in the city, you will need to factor in that traffic to the airport will be much heavier later in the day, so you would be well advised not to leave much later than 3 pm anyway.