Fair question. The school registration process is a real minefield.
Unfortunately, Hornsby House is one of those schools where you really need to register them as soon as they are born! My friend registered her kids for the local private schools when she was pregnant! She didn't even know what the gender was at that stage! It's madness. I don't know what Hornsby's rules are regarding foetuses but they have already closed registration for reception for 2020 and 2021 which would indicate just how oversubscribed they are!
My kids are both at school now and my youngest has just started reception, but I wasn't one of those organised mums that registered their kids at birth. I registered my eldest at around 18 months for a local private school. I called the registrar and they said that we had a slim chance, but hey we got it.
We have since moved schools, but all I would say is that if there is a school you really want to send your kid to, then apply, get to know the registrar if you can, keep in touch and ask if you can go on a waiting list. Having been told that it was super competitive in Wandsworth - if you haven't registered at birth, if you are not in exactly the right catchment (for state) then you haven't got a chance. Well, we got offered 2 private school places (both of which I would have been happy to send my kid to and one we did), and two outstanding state school places. We also got offered a bilingual place at another state primary. I know that my kid would have done well at any of these schools.
My point is, be organised, but don't worry. People move around so much in this area. It is likely that if you don't get what you want straight away, then you will later on. We only had to wait until Year 1 to get into the school we had our heart set on. And you never know, you may be happy with an equally as good alternative that you hadn't thought of!
If you want Hornsby House, get to know the registrar now. If you've missed the deadline for reception, you may get your child in later on.