Despite the no secular schools thing, more and more people in the UK are atheist so it's not particularly effective!
The religiosity or otherwise of a school basically comes down to the head-teacher's personal beliefs. In London most community schools are of the 'nod to god' variety, but if lack of religion is important to you then it's worth checking it out (although a change of head can change that).
RE is the study of faiths rather than the study of faith, and things like Diwali, Eid and so on are also celebrated.
You can withdraw your child from RE and the worship bit, but other than JWs, no-one generally does. DH and I are secular humanists and as atheist as you get, but still like Xmas carols, nativity etc and see it as an important part of our cultural history.
In London traffic an hour isn't often that far. DD goes to a primary that is only about 1.5 miles away but by the time we've walked to the bus stop, taken the bus and walked to school it takes best part of 40 minutes.
Other things to check is the situation regarding bulge classes. A friend of mine has applied for a school that has single form entry (30 children) and had a 150m last admitted distance in 2011. In 2012 they ran a bulge class (60 children) and I have a horrible feeling that they may not get a place this year even if they live on the doorstep. Bulge classes = siblings and with the potential siblings of 60 children in 2012, there may well be no places over in 2014.
Multi-year entry and no previous bulge classes are your friends. The local authority will have this information.
www.southwark.gov.uk/info/200289/primary_school_admissions
The brochure can be downloaded from here and will have all the info on last years allocations. Schools which have become Academies may not have their 'last distance offered' details - you can try calling them individually to find out (although some of them didn't seem to have a clue when I asked last year!)