Most London independent schools assess entrants, and these assessments are incredibly competitive.
The 7+ assessment used by London day preps is notoriously challenging for such young children. It involves tests in English, Maths and Reasoning. Some schools also require participation in group based activities.
If you are considering moving your child at 8+ it is prudent to send them to a pre-prep which only goes up to this age group. The pre-prep will have to ensure that they find all of their pupils a place and therefore prepare them well, so you won't have to spend as much on additional tuition.
8 years old is the traditional age to start at boarding prep school, but many children now start boarding at 9, 10 or even 11. It is really a matter of doing what is right for the child. There are a few notable exceptions, but boarding prep schools usually have an easier admissions process than London day schools. Many use short tests simply to work out the child's academic level.
For a fuller picture come to the Independent Schools Show and speak to the schools in person. Do remember that there are always exceptions to the rules. Independent schools are precisely that independent. Whilst they share much in common entry procedures do vary. These idiosyncrasies can only be fully grasped by talking to each school and discussing their entry requirements.