My appointment in Croydon was at 2:30. We went through the payment process, and were speaking with an immigration officer by 3:00. It only took him 10-15 minutes to approve the visa, but the paperwork is then sent "upstairs" and that's the part that takes 2-3 hours. You just sit there and wait or you can go outside but WARNING. They close the building at 4:00 - if you are inside, you can not leave, and if you are outside (like my husband was in order to put more money in the parking meter) you can not re-enter! So there we were for 2.5 hours: I was inside waiting for my visa and he was outside in the car. Ugh!
When I left Croydon with my visa in hand, at 5:45pm, there were still about 100 people waiting to be seen.
If Birmingham is any quicker, I'd say go for it!
Best wishes.