So, your preference is for all-girls going through from 4 -18, is that right?
Your next preference is for a rigorous academic school like NLCS or Putney High?
I'm guessing you've looked round all these schools and the 2 nurseries you've mentioned? Where do you see her being happiest?
Do you only have a daughter, would be my first question?
Do you only want academics, ie no frills - like Putney High Juniors?
Are you really sure you'd want a child in the same environment right through - what happens if as they mature they need either a bigger, bolder place or it becomes obvious they need a smaller, more nurturing school ... and you've moved house for the wrong school?!?
We were in a similar position to you a few years ago except we had children of both sexes. Despite the fact we live within a 5 min walk of Putney High, we rejected that for our daughter - there were too many things we didn't like on looking round and too many things we felt were missing. (We'll look again at the Seniors when she reaches 11+.) I've no idea about LSH or St Helen's except it's not got the academic reputation of NLCS or the GDST schools - this isn't the be-all, of course and you may have just loved it for your daughter, in which case, it's probably right for you.
When we looked round Thomas's, Battersea, we knew we'd found the right school for our family - it's the most wonderful place - you can feel the excited, zingy atmosphere as you walk in - and it suits my very different children equally well. As they get higher up the school the sheer breadth of things going on is stunning - there's something for every type of learner/nature.
Unlike you, we hadn't really given much thought to the whole SW London nursery/prep school admissions craziness so hadn't registered our first for Thomas's in time ... not a chance of assessment for Reception once we caught on. Except a place came up at the Battersea kindergarten, which guarantees an assessment for Reception, so we took it, really because we wanted to aim for that Reception place. The Kindergarten was a lovely experience for all of mine (the next two followed), mainly play and experimentation, lots of creative activities and things to help learn confidence and independence. There are lots of plays, outside games, sponsored scoots, picnics etc through the year and the children start to develop social skills - serving drinks to grown ups, helping on cake stalls etc - too, which mine all enjoyed. The staff there are lovely, proactive and interested in all the little individuals - children and staff have lovely relationships.
Given that you're Fulham-based, have you completely discounted Glendower? Lots of girls then go on to Put/Wimbledon High or Godolphin & latymer, all very easy by public transport from Fulham. Glendower has a great academic rep plus more going on in extra-curricular staff than Put High Juniors has?
Sorry, long post ... am I pleased not to have to start with this all over again?! Basically, there's so much excellent choice available in London, isn't there?