I also agree that going to a state primary will not preclude you from getting places at the selective secondaries. They are looking at ability and potential over cramming (though few tutors or private preps would want to highlight that, and in some cases I am not sure if they even understand the distinction. I certainly know of incidences where Prep Schools were surprised by who did and who didn't get in to the selective senior schools because doing well in their repetitive testing regime does not necessarily equate to being able to show all the dimensions of ability and personality that the school seek evidence on in the admissions process). Though not perfect they do tend to spot ability even if the pupil has not attended a private prep / years of tutoring, though it helps if they have had some preparation for working in test conditions and with the parts of the curriculum state schools will not have yet covered. All of these schools have a significant number of pupils coming from state schools, especially Hampton because its main entry is 11 and most Prep Schools are geared to 14 + entrance and resist their pupils applying at 11. LEH have around 30% coming from state primaries even with all girls applying at 11. Even, I understand St Pauls Girls', who used to have a pretty miserable record, is also making a determined effort to make its admissions process one that enables those from the state primaries and not just Bute House to be successful.
So if you are choosing a Primary/Prep chose it for whether you think your child given their personality /confidence/maturity / learning styles will be happy there and achieve their potential, not based on what schools they get into at 11. Not all, even the most able pupils respond well to being pushed, whilst there is an issue in the selective 7+ preps with pupils being heavily tutored to get in and then struggling and losing confidence. For example each year there are a couple at LEH Juniors who are advised that the senior school is not for them. The 7+ system is becoming less popular with parents and I know some who have chosen Newlands over that. As well as facilities, extra curricular etc. I would rate it being a local school as especially important. It makes friendships easier and you less grumpy as a result of south west London traffic. And for boys do you want them to be in that school to 14 if that is their policy. I have seen some tall, nearly bearded young men, look distinctly embarrassed by that red taped blazer which looks so cute on a 4 year old! Of course getting a place at a local state primary can actually be harder than getting a place in a Prep!
There is a huge range of private selective, and not so selective, secondaries available to anyone living in Teddington. Children from this area go to, in addition to the ones mentioned, KGS, KCS, St Pauls and St Pauls'Girls, Godolphin and Latymer, Latymer, Putney High, Surbiton High, Reeds, Claremont Fan. Slightly harder if it is a boy but still you will have many options from which to chose the school that will suit your child best at 11.