Hi - I am actually a birth doula at Chelsea and Westminster hospital but I can really recommend Maggie Vaughan www.alacartedoula.co.uk who also works here part time. Good luck with your search and pregnancy.
I find this an interesting debate. I have worked at both as a doula and I find the care to be a very similar standard; they can both be absolutely fantastic and then sadly just very occasionally both a little lacking!! At the end of day I think it boils down to where you feel most comfortable (safe)...