My wife and I had to choose among three of the schools you listed (as well as a couple of others) earlier this year, as did many parents at daughter's school. There's a theory in economics called revealed preference, which boils down to: ignore what people say, look at what they do. As an aside, it's interesting that a large proportion of state school heads and left wing politicians send their children to private schools or selective state schools whose intakes broadly match those in the independent sector, whatever they may preach.
There are around 36 girls in my daughter's year. Of about a dozen girls at the top of the year, 10 got offers from North London, 6 from St Pauls and 5 from Henrietta Barnet, most of whom also got offers at City and/or South Hampstead. Overall, the school got 17 offers from City and 26 from South Hampstead.
Of the six with offers from St Pauls, five are going and one turned it down for Henrietta Barnet. Three of the girls going to St Paul's got offers from HB that they turned down. Of the 10 with offers from NLCS--six were awarded scholarships--just three are going, only one of whom got a scholarship.
Of the girls with places at both City and South Hampstead, almost all have opted for City. Why, I don't know. Had we been choosing between the two schools, we would have opted for SHHS, which ranks higher on the exam tables even though the best sixth form girls tend to be poached by Westminster (six this year, I think).
Our very difficult decision was between SPGS and Henrietta Barnet. In the end, we opted for SPGS for a number of not entirely rational reasons. One, we could afford it, and it has excellent facilities and teaching staff, though it was painful to turn down a free place at an excellent school. Two, many of the successful women we know went to St Paul's (this might be because Paulinas let you know pretty quickly where they went to school or simply that we remember because we already know the school by reputation). Three, my daughter's school routinely sends its brightest and nicest girls to St Pauls--it is the head's strong preference where there is a choice--and it's hard to ignore precedent. It helps that SPGS sends more than half of each year to Oxbridge.
Of all these schools, I felt that NLCS does the hardest sell. I didn't like how we were marketed to ever since we looked at the junior school as a possibility for my daughter all those years ago. The "we are the best" sales pitch grates. I also found the manicuredness of the place offputting, more like a business school than a secondary. Faced with a choice between NLCS and SHHS, we'd have probably gone for SHHS (as, indeed, a friend's daughter who's at another primary school, did). I think SHHS is as underrated as we felt NLCS to be overrated. I suspect, deep down, that SPGS is also overrated, but we've made the choice, so we'll lump it. We immediately knew we preferred HB to NLCS.
None of this tells you much about the schools themselves. You can go around the libraries and classrooms and playgrounds and get a feel for yourself. As far as I can tell, they all teach bright girls all the stuff they need to go to university.