Reception is the equiv of your Y1 isn't it? It isn't like Kindergarten - there is a lot of free play time but it is far more structured. Also, any good school will easily take all that in their stride. They are used to a vastly different level of ability at KS1 and will differentiate accordingly. As for size - well there's a huge difference in size in my son's class but that's never made a difference.
If you want her to go to a state school I would start with the area that you want to move to. Find all the nearest schools, take a look at the Ofsted report for each one (although that by no means paints the whole picture) and look at the school's website for admissions criteria and also the local authority (in the case of Dulwich Village, Herne Hill, East Dulwich and West Dulwich this is Southwark).
This is the school finder on the government's website:
schoolsfinder.direct.gov.uk/schoolsfinder
To do a quick look up on all the stats and ofsted reports for a school look at its dashboard - enter the school's name in this finder:
dashboard.ofsted.gov.uk/
This is a link to the government's website on finding schools:
www.gov.uk/schools-admissions/choosing-schools
This is a link to Southwark's web page for school admissions:
www.southwark.gov.uk/info/200289/primary_school_admissions
When you arrive and have a residential address in Dulwich (assuming here!) you will put in an application to Southwark - but the school does not have to be in Southwark. They can be in any London Borough - you just need to put in one application. For example, if you were on, say the Croxted Road side of Dulwich Village you might list Dulwich Infants (which is in Southwark) and Rosendale (which is in Lambeth).
I would suggest that you call around, and see if school's have a vacancy. You might need to go on the waiting list for schools - often people don't take up a place but don't notify the school. When a place comes free it is allocated using the same admissions criteria, not who was put on the waiting list first. So if you are closest to the school, for example, you will get a place.
You might want to look at an improving school. Langbourne, which I believe has been renamed Dulwich Wood, has had a lot of money and time invested in it by Southwark recently. It is in a lovely setting, and has something many Lambeth schools lack - lots of outdoor space. It's not quite so handy to Dulwich village, but is still walkable, and easily less than 5 mins drive.
On the West Dulwich side you have Rosendale and Elm Wood, both outstanding schools. Elm Wood is also in the process of expanding so there's lots of building work going on but it has gone up to 2 classes per year (what they call a 3 form entry here). And Rosendale as 3 classes I believe. I'm afraid I don't know the schools in Herne Hill and East Dulwich as well but I think they're all good schools.
Also state schools in the UK can be religious - Dulwich Infants is a Church of England School. St Anthony's, which is lovely and has an Ofsted rating of Good, is Catholic.
If you decide to go private then there's lots of options in Dulwich, and you can find them using the same Government school finder. Note that in the UK an 'Independent' school is a private or fee paying school.
Finally, although a child has to start school at five by law, this doesn't have to be a school. All you have to do in the UK is let the local authority know that you are going to home educate. There's no application, assessments, or follow ups. At any point you can re-enter the school system when you want to, by putting in an application with the local authority.
Note that if you are offered a school place and turn it down, the local authority is considered to have fulfilled its obligation to provide an educational opportunity, and does not have to offer you another place (and probably won't).
All this sounds stressful but it isn't really. Oh ok it is! But you can come here and ask as many questions as you like. We have two really lovely regulars, PH47Bridge and Admissions who know everything there is about applying for a school place and have endless amounts of patience at helping people. I'm always amazed at their kindness in donating so much time here.