The fact that this issue has arisen at all is ridiculous and the policy of First Bus ("first come first served") is very surprising.
As has been said, today's buses have been designed to accommodate wheelchair users. They were not designed to accommodate unfolded buggies. Wheelchair users are protected by law whereas unfolded buggy users (and indeed ordinary passengers) are not. The space is a convenience for buggy users but not a right. London buses have a clear policy stating that wheelchair users take priority over buggy users. Each bus has a sign to that effect and TfL's website states this quite clearly:
"When travelling with a buggy, feel free to use the wheelchair space if it's available. If a wheelchair user wants to board the bus, the driver might ask you to share the space, move your buggy to another part of the bus, or, if necessary, fold it. Wheelchair users have priority over anyone else in this area as it is the only place they can travel safely. "
Note the term "Wheelchair users have priority over anyone else in this area..". It includes ordinary standing passengers and I have been on buses where the driver has told passengers standing in the wheelchair area to move upstairs or get off when he has arrived at a stop where a wheelchair user was waiting to board.