It would depend on the specific private school.
There are private schools for everyone. Obviously if you are targeting a highly selective school, then it's going to be more difficult to get in than a private school which takes a lot of children with SENs.
What area of London are you in?
I know a parent at an Ofsted outstanding London primary and she was very disappointed because they did literally nothing to support the 11+ process (religious school so they don't care, even though as I understand it he is not eligible to go on to their linked senior school), and he failed his 11+'s although is IMO very bright.
From what I've seen of the boys the private school 11+ exams are hugely competitive in London, although obviously there are less sought after schools where you could get in easily.
I don't think you actually need to tutor at all, I would say my son did a lot of verbal reasoning and non-verbal reasoning tests at school, but these are straight forward to do at home if you are disciplined, and one-to-one work as a parent is very effective. For maths, my son is very able and we did nothing at all. For English a tutor probably would have helped, as he is weak in this area, and really needed one-to-one professional help, but he appears to have got through on his maths.
No school will fix big problems with your child's abilities - this requires one-to-one help, what you get from the prep school is:
- lots of exposure to the 11+ tests (but you can do this at home)
- a good, detailed reference (the senior school should write to your primary school - there is no guarantee that your head will reply)
- limited interview prep (DS was told some of the questions asked of previous candidates, given tips such as sit up straight, tuck your shirt in and so on) - again, this is not really rocket science.
- good general academic progress
You will be assessed, probably (check with the schools), in January of Y6, essentially on maths, English and probably some silly IQ tests. If you know that your daughter's maths or English are poor, then by all means get her a tutor now. But if they aren't, then I would leave it till Y5 when you have visited the schools in the Autumn Term, and then you will have a very solid four terms to prepare the VR/NVR nonsense, which is much longer than I spent with my son.