Word of mouth is good, and booking a year before you want to start tutoring should give you access to the better ones.
Most expensive isn’t always best. Most super selective state grammars, and popular independent day schools that take a good proportion of local people, have a number of very experienced and successful tutors who have been prepping kids for those specific exams for many many years. They don’t always charge the top rates, but they probably won’t be cheap.
Get to know a few parents in a couple or three years ahead of your kid and ask for recommendations. Parents can be rather tight-lipped about top tutors but are often more open when they are past that phase. If you know people with kids in the selective school who got in couple of years ago, they will often know the ‘known’ tutors who are still working. They can also tell you the going local rates.
We are just inside London boroughs and paid £40 per week 1-2-1 4 years ago. That was for a top local tutor who had been a primary school teacher and was a parent of 3 kids who had gone to the local selective school and who tutored 5 kids a year for some pin money, but wasn’t in it for big bucks. She didn’t tutor during most holidays, except the holiday just before the exam, and if there was an odd slot you couldn’t do, she wouldn’t charge. She was well known locally, especially by the local state school whose parents used her to try to even-up the ground with the local prep school applicants. She was excellent in giving parents an initial assessment for £50 and telling them if someone really looked like they had no real hope, and was succcessul in getting lots of scholarships. It was all word of mouth and she would take your name from Christmas of Yr4 onwards, and would confirm a place and tutor from January of Yr5, after the previous cohort had sat their exams. She was never willing to tutor anyone for more than a year.
Definitley ask around locally for names and prices. If you know any local teachers as friends, ask them too - they often know people who do it.
Also know that with some time and dedication and purchase of some books, most parents who have a reasonable level of education themselves can do it.