I'm not SW London, but I'm ex-London and I clicked on here to say what others already have. You may not have much choice in schools if you are using state schools. You need to find out what your LA's admission criteria are and what their effective catchment area is. Then work out which schools you might have a realistic chance of getting into most years, including accounting for any faith criteria, etc. A change to straight line routes is actually probably a good thing. The catchment will shrink, but probably so will your distance, and it is much easier to do some rough calculations on your own distances yourself using this basis.
The general school stuff isn't that different. Both countries run their school years to the long summer break, it's just that summer is a different point in the year. And the schooling years aren't that different. If I tabulate ours:
Nursery/pre-school (age 3-4) - optional
Reception (age 4-5) - technically optional until the term after your child turns 5, though few people opt out
Year 1-6 (age 5-11) - compulsory
Year 7-11 (age 11-16)- compulsory
Year 12-13 (age 16-18) - optional
Reception to year 6 will be in one school (or, alternatively, reception to year 2 in an infant school, then years 3-6 in a junior school) so reception seems more like 'school proper', but in fact it is much more like a continuation of nursery/pre-school. Very similar these days to the idea of kindergarten in many countries. The only real difference is that mandatory schooling starts one year earlier and stops one year later.
As an April birthday, your daughter (like mine) will actually be one of the younger ones in the year. the oldest children are those born Sept-Christmas. DD1 starts in September and she is so ready for school.