From my experience, when we moved back to Coventry we went to the very local school we wanted our daughter to attend and they said they were full and no spaces so no chance. We then contacted the Council and there was one school close by with spaces so we were able to get her into that school. When it came to Secondary school it was again a bit of pot luck, but we did get offered a space in our first choice and really pleased as it's an excellent school in most cases. My brother relocated back to the UK and at one stage my niece had to travel a few miles to school, by bus or lifts until a space became available in a more local school, she still has to travel quite some way though.
I would make contact with the councils and schools you are interested in to see if they have spaces in the correct year group, and how quickly they tend to fill, before making the decision. Rightmove.co.uk also have a school finder tab and allow you to see information on the schools such as Ofsted reports and might be a handle tool for you too.
Thinking there might be some schemes to further assist on the property front I just did a search for houses for a pound, but looks like Liverpool have stopped taking applications and Stoke require criteria that might not be able to meet. If you phone the councils in the areas you are interested in, might also be worth asking if they run any schemes that might help you on the property front too, you never know.