Giving birth in the UK (London)

Hello! Right after I got expatriated, I found out I was pregnant. I'm used to better health systems/care than the common NHS way for childbirth, where you get assigned to a hospital. I have changed already , as the one they wanted to send me to, is really not according to the standards I wished for. I also found out that my private health insurance does not cover child birth. Does anyone have a recommendation about which are the best hospitals in North London, without paying a huge private fee? Standalone child birth centres are also not an option for me, as they are to risky if a transfer to a hospital in case of an emergency is needed. Thanks a lot! -- Yanna