For state schools, you must apply through the local authority in which you are resident, but can apply for schools in other local authorities as well as your own.
The schools themselves - or the school's 'maintaining' LA on its behalf - rank your application according to the oversubscription criteria in their individual admissions policies (which must be determined and set out on the school's website 18 months in advance of the September in which the intake will start at the school. The information as to whether your DC is ranked high enough to be offered a place is fed back to your LA via the LA in which the school is situated. Your LA collates the responses from all the schools which you named as preferences and you are allocated the school which you ranked highest which is saying Yes.
(If none of your preferences say Yes, your own LA must offer the nearest of its schools which has a place available, after all the on-time applicants have been offered).
You need to read the information in the schools' individual admissions policies with regard to how they rank applicants (but be aware that these can change, subject to the school having consulted with the wider community for a 6 week period between October and January essentially two years before the September in which that intake will start at the school).
The operative date for being 'provably' resident at the address from which you are making the application is generally the national deadline submission of the Common Application Form, which is 31st October; some LAs have a slightly later date for movers in, but equally, some individual schools may stipulate longer periods of residence (although there have been some successful legal challenges with regard to this).
You will have to put your DC's current primary school on the CAF and landing them with such a long daily commute, rather than applying for a primary place near to your new, apparently permanent, home address may raise a red flag that the the address given may not actually be where the child is living.
Every LA should publish guidance for parents re completing the CAF - which will include a warning about what will happen if it is found that a place has been allocated based on information given by the parent that is found to be false.