Hi LuluT,
I did not have the chance to read through all the posts, but wanted to say to be aware that the school system is very hard to navigate (think Manhattan, or harder), unless you come at the reception time. As opposed to the US, here living in the school catchment (even across the street from the school) does not guarantee a place at that school, unless you register in advance for year 1, 7 etc. As you have three girls, chances are you will not have a good place for at least one of them (not to say that in fact you may get a very bad school). Also, be prepared for the high student/teacher ratio, that may affect a bright student (although in some schools children are placed in ability groups; but you get the kind of attention you get in a school ranked 10 or 9/10 in the US only in a private school here).
Just try to make some phone calls to asses the chances to get a place at a school of choice (and do not wait until July, as we did)...you'll be surprised.
What I would do based on our experience (and pain) after relocating to the UK? I would live somewhere outside London where the rent is lower and would consider private schools for at least one of the daughters who will not get a place at a decent school.
I am sorry if this sounds a little pessimistic, but we went through a very painful experience (we had no choice but enrol our bright child in a "good" primary school which later was downgraded to "needs improvement"). It was so painful that later, even if we had the chance to have him in an "outstanding" secondary, we considered private school (indeed, a very good and famous one).
Good luck and I hope things will go better for you.