We applied last year for places for dd1 in a range of pushy London schools, including City of London Girls and South Hampstead High as well as state grammar schools. She did have a year or so of extra teaching, but not specifically exam cramming, more all-round English and Maths tuition to make up for the short-comings of the curriculum in her state primary school.
She was offered places in all the schools she applied to, including one scholarship. She is naturally quite academic, but there is a big transfer of kids from state primaries to the London day schools, though some far more than others. I disagree with bossykate about the prep school kids transferring at 13, though -- that may be the case at country or co-ed preps, but most of the London girls' preps stop at 11, so state school children are competing with the privately-educated ones. Outside London, though, and particularly for boys, transferring at 11+ might be an advantage as privately-educated boys are more likely to transfer at 13.
You should be able to find out where your preferred school's intake comes from -- some London schools are quite happy to supply on request a list of local schools, state and private, from which they have accepted children over the past, say, 5 years.
Find out, too, what the secondary school's entrance exam consists of. Verbal and non-verbal reasoning papers are designed to measure innate ability, so as long as your child has had a chance to practise the appropriate tasks (you can buy workbooks in eg. WHSmiths) you should be fine. If the exam consists of traditional written exams in English and Maths then some extra tuition might not go amiss. IME the most likely gaps in a primary-educated child's repertoire are in the area of essay-writing (dd1 did virtually no extended writing in school of the type required in private-school entrance exams), more sophisticated reading-comprehensions (the Y6 SATS ones are pretty basic and idiot-proof) and more advanced maths (eg fractions beyond just the basics, simple expressions and equations and more complex problem-solving).
I would personally be wary of tutoring a child to the point where they just scrape into a school, though. Dd1 is happily ensconced at secondary now and loving it, but I wouldn't send ds to the boys' equivalent of her school because he's bright but not exceptionally so, and I would hate for him to struggle. I think aiming to have your child in a school where they can be in the top third, or possibly top half of the ability range is ideal -- I would be reluctant to send a child to a school where they would be in the bottom 30%, as I hate the idea of them having to stress about keeping up.