Perfect sense.
I'm glad you're visiting and getting to make your own mind up, good for you!
Big mix, and by that I mean there are a fair few on free school lunch, increasing numbers of middle class/professional families, and one little boy I remember having a gigantic birthday cake delivered from Harrods, big enough for his whole class to eat twice over 
So...a big mix. The whole range.
I come from the other end of Barnes daily, nearer the pond, and my dds have had no problems at all - far from it. Lowther is hot on inclusion, whatever the kid's background. I haven't had any problems either, and have so many good friends amongst the mums there now - both from the Estate and not.
Both dds have friends who have gone/are going to independent schools at 7+ and/or 11+, and they may well do too at 11+...depending on state of local schools at that point.
But, do go and see - its not for everyone, I'm sure. It does depend on your priorities for a primary. I looked at two indies, two state primaries, and chose Lowther - but had many raised eyebrows at the time (four years ago)! No regrets, though.