Hello Emily,
For networking w/ expat women from all over, this is a great club and how I met many friends over there:
KCWC - The Kensington and Chelsea Women’s Club
they have various groups & social activities for all ages and backgrounds, working women to parenting (which has numerous groups all on its own)
Kensington, S. Kensington, Chelsea, and Fulham are good neighborhoods, or Burroughs as they say, for you to search for houses or flats.
Best of luck & let me know here if you need any specific info on restaurants, good local websites, etc.
Cheers,