We moved mid-year last year, so looked at loads of local schools (we're slightly further north than you, though). The fruits of my researches were:
For Eleanor Palmer you'd need to live in a tent in the playground, as far as I can work out. Headteacher was unbelievably rude to me when I phoned, so I went right off the place.
I thought Hawley kids moved en bloc to somewhere like Torriano, or may be I invented that? Check the admissions policies of other local schools, I have a feeling some prioritise Hawley kids at junior level. Or ask Hawley where their kids go. Torriano is quite a big school, not sure how far the catchment area goes. Liberal, no uniform, blahdiblah.
Don't know about kentish town, but there seem to be not a few muslim etc kids, so can't all be churchgoers.
One of my dc was offered a place at St Patrick's which seems nice enough.
Non-church schools are not allowed to prioritise nursery kids for Reception places, and indeed my dd2 has just failed to get a reception place at the school she's currently at (we must have been micromillimetres outside the catchment by my reckoning).
Finally (and we've just been stung by this) don't underestimate the no. of people who rent in catchment for 6 months before their oldest child is due to start school, and then move back to wherever. We live 5 mins walk from the school dd2 has just failed to get into (2-form entry, ie. 60 kids per year group), and the no. of people who drive past us to deliver their kids as we walk the few hundred yards from our house to the school is unbelievable.