As most nursery schools are small, it's advisable to register your child as early as possible if you want to be on top of the waiting list.
But don't be disheartened if you don't get your first choice straight away. Stay in contact with the school, as a lot can change in the year between offers being made and the dates when nursery starts (particularly in London where families may relocate or decide to decline a place, freeing up places).
All nurseries differ in their entry requirements, timings and approach. Some nurseries, like prep schools, will only have a September intake; others will have January and September intakes; some even termly intakes. At this early age, a few months can make a huge difference in a child's development.
Entry requirements at ages 3 or 4 vary considerably but nearly all schools now have some form of assessment before they offer a place. This can vary from a written report from a nursery, to a mini-assessment day complete with an individual chat and observation as part of a group. The more academic the school, the more likely they are to test each applicant.
Schools aren't expecting your child to be reading, writing and doing long division aged 3. Rather, they are looking for children that enjoy learning, retain knowledge, show signs of being confident and independent and can carry out instruction.