I've done a few CPT- GBE flights in and out of Cape Town so the planes are a LOT smaller and fewer passengers but no immigration has taken more than 5 to 10min. They seem to be pretty jacked up and have a decent amount of staff manning the immigration desks. Cant say for sure on the bigger BA flights but shouldn't be an issue.

Edit: I am not sure if they are using the new fingerprint scanning and photo procedure yet. This might delay the process a bit.

I dont think there is any "pay per use" showers.