Presumably you know your new address so can see which schools are nearest. I am also assuming that you have done some research into the schools near the new house. I don't know if you have had a chance to visit possible schools but obviously that would be a good idea before you make an application.
I think most LA will accept requests for school places once you have notification of exchange and a date for completion. They might require proof of this from your solicitor.
Some schools will deal with their own admissions, others do it through the LA. I would contact schools you have earmarked and find out what they do, you can also ask them what the situation is re available places in relevant years groups/waiting lists. THe LA should also be able to provide this information about all schools in the authority.
Depending on what the school says you either apply through the LA for in year transfers or directly to the school. Bear in mind though that most LA/schools will expect you to take an offered place within a week or ten days, they will not usually let you hang on to it if something goes wrong with your move.
There may not be places for both children in the same school you want bearing in mind that one of your children will come under the 30 pupil infant class size. If this happens you could either accept the places offered in different schools and put both children on the waiting list for the other ones school, or ask the authority if there are other schools that do have places for both . If your younger child is offered a place it might also be worth appealing for a place for your older child. Sadly an appeal the other way round is unlikely to succeed.
Good luck, it could be that you are fortunate and the school you want has spaces for both, but you need to be prepared for some compromise/waiting list angst/two children different schools organisational nightmare.