It isn't the same at all private hospitals, and it can also vary depending on whether you planned to have a Caesarian (elective) or end up needing one for emergency reasons. If you look up comments on places like mumsnet, you will see people quoting figures between £9000 and £13,000 - the last figure seemed a lot higher than the average though, so i think it must have been an exception. It can also be much cheaper if you have a private operation at an NHS hospital, but this option many not be available everywhere.

To give you an idea, you can see a list of one hospital's charges here. As you can see, the prices vary according to the exact type of operation required, type of accommodation etc. To get an accurate estimate, you would probably need to contact a particular hospital or go on their website and look up their charges.