Some schools are non-selective, i.e. first come first serve (e.g. Hornsby House, Finton, Eaton House the Manor, Hurlingham Prep, etc)- for these you have to register as soon after birth as possible. The phenomenon of people sending letters of application on the day of birth is not an exaggeration.
Some do an assessment, but only of children who are on registration list, i.e. they will not assess 500-1000 so they keep a registrtion list of say 200. For these, you also have to register soon after birth, e.g.Thomas', this will not guarantee a place,but will guarantee an assessment.
You can also view schools now, don't need to register. Some schools you won't like the look/feel of and then no point wasting the registration fee. As a matter of fact, I would suggest doing it now, as you will have your hands totally full once the baby is born.
Also, you need to start looking at the nurseries - for those, you really do need to register straight after birth. Eaton House and Thomas' apparently have nice ones, but plenty of local options. But again, you need to go and look NOW while you have the time and the energy. Quite exhausing actually, this looking around stuff