My DD is Yr6 at a very good state primary and is currently applying to independent secondaries. We will find out in 2 weeks how successful she has been, though I'm cautiously optimistic.
My 2ps worth - LondonMum's comment does not reflect our experience and frankly that attitude may be a reason to be wary of doing private the whole way through. Whilst DD will probably go private, she is well aware that some of the best kids in her class, from lovely families (both mc and non mc) will be going to the local comprehensive. She's also aware that while she may be better academically than some kids, many have other talents (art, sport, humour) where she is not even close. Even if she goes private for secondary, she'll never have that feeling of 'letting her kids down' if she later sends her own children state.
The tutoring has been fine. She's done roughly an hour most weekdays for yr 5 and the first term of year 6. Whilst of course this interferes with family time, my understanding from neighbours with kids at prep schools is that they normally get at least one hour homework in those years, and that the vast majority have private tutors on top.
I also expect the stress is worse at prep schools. In DD"s class she is in a minority, so it's not a subject that gets discussed much at the school gate. The children are also largely unaware of the ranking of different schools, which I doubt is the case at a prep.
However, that said, I had a look at Kensington Prep's offers list and it is impressive - a quarter to St Paul's. My impression is that it is very hard to get into the absolute top schools and a prep gives the edge there, but for a more mid-ranking senior school - say Putney High - a prep doesn't add much. So while many preps may not add much beyond an outstanding primary, that particular prep may be an exception.