We've just gone through the entire 11+ process with our eldest son. He's a very bright boy who wasn't being stretched enough in his regular south-London public school, and he approached us last summer to say "daddy, can I do the 11+ and go to a great school please?". His mother and I advised him that he'd have to work hard at it, and he replied that he was determined to go through with it. So this was not something we forced on him at all.
We applied to Westminster, King's, Latymer and UCS, and started working with him on preparing for the written exams 5 months before they took place. His school put on extra lessons, I coached him on maths every day, and we got the help of a tutor twice a week in the last three months for English comprehension and creative writing. By the end he was in extraordinary shape: he could write exceptionally well (the tutor said she'd never seen a 10-year-old write like that), and he was able to do complex simultaneous equation problems in maths without batting an eyelid.
He sat the 4 sets of written exams, and got invited to interview by all 4 schools.
We then coached him for interview. He's a very confident and talkative chap, so this was a doddle. The headmaster of his current school personally trained him on interviewing, and told us "he should have no problems at all". Our lad came out of the interviews brimming confident and happy - he felt he had "aced" them. (UCS, for example, ran a "mock class" for groups of ten boys at a time. They projected an image of Marcel Duchamp's "urinal", and asked "is this art?". Our boy says everyone was stone silent except for him, who replied "well, it starts with an idea, so yes it is art").
Despite all of this, he ended up with just one school offering him a place (and it was not UCS). He got rejected outright by 3 of the 4.
All of this leads me to conclude:
(1) Getting into the top private day schools is incredibly tough. No matter how bright your kid, nor how much you prepare, there is no guarantee you will get in anywhere.
(2) It doesn't matter how smart your kid is or how well they do at the written exams - at the end of the day it just comes down to whether the school "likes" the child. That's totally random - you can't prepare for it. The school that offered him a place featured the interviewer spending 50% longer than the allocated time with our son, caught in a deep discussion about the engineering challenges involved in building a flying car (our son's pet project). The two obviously "clicked", and he got an offer. It is disconcerting that "clicking" is what in the end makes the difference between an "aye" or a "nay" - but this is what this process boils down to.
So: if you're going to go for the top schools (and unless you have a child that is freakishly intelligent), plan on spending 4-5 months of hard work every day preparing them for all subjects. That'll get them past the first filter - ie the written exams. Then coach them as much as you like on interview technique, but expect to be presented with a random set of outcomes. In this game, it becomes about numbers- so apply to at least 4 schools of differing styles to avoid ending up with nothing.