I'm with Mavis. A couple on your list that My dog (if I had one) would be spared. But which are very sought after by others.
I would register early. Not least it should that you are settled in London and so are more likely to last the duration. But also shows a consistent interest in the school. Most of These schools are selective.
If you then have a choice, start with the child and consider which school is most likely to help them develop. For example confident children may thrive in a bigger busy schools, quieter children may prefer something more nurturing.
For girls preps as well as looking at the children outside school, perhaps go into a nearby cafe on a Friday morning in term time and eavesdrop on the large group of mums who will inevitably be gathered there. Is it a group you would feel happy to be part of? I say this because there are some super competitive mothers out there, and this rubs off on children. We are almost at the end, and this is the aspect of raising children in London I have enjoyed least.
Education is a long process. If possible try to avoid seeing it as a race. Go to the right prep, and SPGS and Ivy follow naturally. Some manage it but there have also been some awful casualties. All the schools you mention can get the right child to the right place. Find the school that suits your daughter. Then at 11+ do the same. Children develop at different rates and have a range of talents. SPGS will be the right school for some. Fairley House for others. What is important is that kids emerge with resilience and emotional intelligence, their talents nurtured and their self esteem intact.
I also agree about adding Ken Prep to the list. Perhaps Newton Prep if you can get there. Both produce happy confident children.