As someone moving into an area with a child needing what is called an 'in year' place (ie not one that is at the normal point of entry to school), you are in a tricky situation.
Even moving next door to your desired school may not give you a place there, if the year group is already full and no-one moves.
On the other hand, if you live too far away from your desired school, then you won't be top of the waiting list if a place DOES come up.
Furthermore, it is very hard to 'hedge your bets' by living reasonably close to several schools, just because of the way school 'intake areas' work.
The only way you can be certain of getting a specific school in your situation is to go private.
To maximise your chances of a decent state place, then moving close to a good school which typically has a highish turnover could be helpful (a school which always has places is unlikely to be a good one). Asking the council the specific question 'Please could you give me the number of in-year places given in reception, year 1 or year 2 at the following 4 schools over the last 5 years?' will give you an idea of turnover.
Or you can just move, apply, and hope. On the positive side, if you are in the country, and a place arises anywhere that isn't immediately filled from the waiting list / if you are the top of the waiting list, that place is yours - you will have to take up the place almost immediately. Also, you will have to be found a school place ... somewhere ...
On the negative side, you will have to decide what to do if the place offered is undesirable - do you take that place and stay on waiting lists for all your preferred schools, or home educate temporarily until a place comes up? If you do decide on temporary home education, it will need to be a valid option for you until the start of year 3. At that point, class size limits are relaxed, so while a school may refuse to admit a 31st pupil into Y1 using Infant Class Size law, they may well accept a 31st at Y3 if you appeal.
The thing the process lacks is certainty for you, which is a real pain when you're moving from abroad. As I say, the only certainty would be a private school with a vacancy.