Congratulations on your pregnancy. I can't comment on Thomas's as I had my little one at C&W via planned C-Section but I think the modus operandi vis a vis private rooms is similar, they are on a FCFS basis, as well as priority of need - often women with multiple birth situations or women whose babies have been taken away to the NICU might understandably get priority.
For example, I had a private room at C&W and in the middle of the night they came into me extremely apologetically asking if I would mind moving to the ward due to a medical emergency (but that I didn't have to if I didn't want to). I was very happy to do so as it happens. When I gave birth at C&W, private rooms were around the £300 mark but from what I hear they are more than double that now presumably to reduce demand which always far exceeded supply.