Now that you still have time - do some research about the school's near you. I found Good Schools Guide (online, you can subscribe for 1month) - very informative.
If you know the gender of your child - you can focus on boys or girls schools. If you are waiting to find out - the you need to research both types, and co-ed.
Here in the Kensington and Chelsea area - there are some "first-come-first-served" and some "by assessment" schools. With the first type - it makes sence to register as soon as possible. You only have to pay a registration fee - so it gives you an option. With some of those - Wetherby - it might help if you call from the hospital - (I know, crazy!) - but it is VERY thought after...
For other "assessment" schools - it helps to register in the first 6mo.
Knowing how busy people get after the birth of the child - it's better to research/prepare the applications before the birth. Ours were ready - missing the DOB and name - before. Then my husband took care of them while I was in the hospital.
It sounds crazy - but, unfortunately, it is what it is around here. And do trust me - once the baby arrives, last thing you'll want to do in the first year (or two) is research schools. So - why not prepare in advance...
With my first child we registered at 2 non-selective, and 4 selective schools. We wanted to have a range of options.
Same is true for the nurseries. Only you don't need to register with as many. Just a couple you like. And it helps to tour them while you are pregnant - shows you are really keen.