I've not heard good things about St Martins or St Philips from various people.
They were not choices of schools for us, so cannot comment much, other than what people have said.
I don't live in that area where St Martins is.
My child goes to Combe Down Primary and we live in Claverton Down which is close to Combe Down and Combe Down Primary is the closest to us, so generally if you are living in Combe Down or Claverton Down your child has more or a chance of getting in to Combe Down Primary.
The Bath University is in the Claverton Down area / along Claverton Road.
I do know of a family near us who had applied for their child to attend Widcombe Infants which is also quite close to Claverton Down (not as close as Combe Down) and the child was given a place. Normally this doesn't happen as with Widcombe it really depends on how close you are to that school which is also quite over-subscribed. Widcombe would've been our first choice but because of the distance in terms of the catchment area we didn't get a place when we had applied but as I mentioned this family did the year after and it just so happened that they had spaces when they had applied.
Our choice for Widcombe was that it came across as a very nice school when we had a visit (grounds and head teacher /atmosphere) and easy to get to as well, but we knew that we would get Combe Down based on where we live.
It's quick and easy to get to Combe Down Primary school from where we are and we live in a nice area(my opinion).
You'll probably prefer Combe Down Primary for your one child.
I don't know the areas like Odd Down but there are quite a few children who live in Odd Down who have places in Combe Down Primary, so if you really prefer Combe Down Primary then perhaps you should focus on those two areas for living.
With regard to Weston - I can't comment as I've not lived there.
I'd add that from just general knowledge of what people say, I think from all the schools in this area - St Martins or St Philips and Combe Down Primary most would say you'd be much better off with places at Combe Down Primary.
These comments are based purely on what I've heard from various people who live in the Odd Down area.
Good luck.