I am not sure what teatime is getting at but it may be the link system which was stopped three or so years ago. Entrance to the state Senior Schools, apart from St Richard Reynolds and the foundation places at Christ's which have faith related criteria, are purely on distance from the school gate (apart from looked after and statemented children and any sibling policy). There are no formal catchments so that distance is purely driven by demand, which is intense for the outstanding comprehensives, Orleans, Waldegrave and Teddington. Turing not yet had an OFSTED but popular too. They do however provide allocation maps from the previous year which are a guide. Demand is increasing but the new school at Egerton Road should alleviate that in the near future. I have never heard of any of the state schools having a grammar test as a basis for the admissions criteria in this borough, perhaps the tests to determine ESL status but this would not be an admissions criteria? The fact that 30% from this borough go private for secondary education (and many move away) is partly the result of the pressure on good state places, it is not just affluence.
The situation with regard to entry to Primary Schools was even more intense in both Twickenham and Teddington, even given the expansion of existing schools which had reached the limit of feasibility. Two new Free Schools have come online and more in the pipeline so hopefully parents are no longer at risk of finding themselves allocated only a school that is inaccessible or without a place allocated until, in one year, Christmas. Part of the problem is the number of primary schools with faith related admissions criteria, so sitting in pews has been a common way to avoid the stress. The schools in Twickenham and Teddington are very successful and the Council has long had a strategy of blaming the problem on parents moving into the area to access the schools and it suited them to deter parents by demand exceeding supply. It therefore requires a bit of strategic planning in terms of what roads you move to be sure of accessing the best schools.
Entry to Tiffin is very competitive, over 2000 applicants and you have to score in the top 3% on measures of verbal and non verbal ability. Tests in English and Maths have now been added. There are endless threads on Tiffin and tutoring including whether you would want your child to go there in the first place given the tutoring culture. For the selective private schools the admissions process is more rounded with examinations designed to measure the pupil's potential, as well as interviews and references. In general the more selective the more they will be looking for intellectual ability (not crammed knowledge). Tutors will tell you that tutoring is essential for entry to both and many anxious parents have been engaged in an arms race starting earlier and earlier with greater intensity but bright children can and do get into these schools without it, though working with your child at home to fill the gaps in the Year 6 curriculum that has not been covered yet and prepare them for examination conditions on top of all the usual stuff you would do to stimulate your child to think and read is probably a minimum. There are more private school options for girls than boys.
I do think this is one of the most stressful parts of London in terms of state school entry and I have friends in South and South East London who are shocked by what parents have been put through. Hopefully as I say the new schools coming on stream on this side of the river will improve things.