There are so many 'rules' to this.
Call your admission authority where the house is situated. You may want current information, which may be indicative of the schools which may have places. Of course this changes. Where I live, there are places at 'undesirable schools'. Anyone can call up and get that information.
So if you own the house, you are moving back to that exact house, then get an application form online from your council (online even) and apply for the school you want. It may be a rule that you have to be all resident in the house, but if you are talking a matter of weeks, and it is your house, it's not going to change distances to school and when going on waiting lists, the time you go on it can be important. Yes, this may be a minor rule break, but if you are living there, I would apply asap regardless if the children are two weeks behind you.
If they don't have spaces, then the council will tell you where does have spaces. In fact, they will tell you NOW where has places for the years of your children.
After applying, you can go on waiting lists etc.
Once you get either child into an acceptable school, the other will likely go to the top of the waiting list. You can put them into a different school whilst still waiting, just keep on top of the admission people and the school office to ensure they are aware you wish your child to remain on the wait list. You could put each child on a wait list at the other school too.
As to whether the children need to be living here... If you are talking months, then I would wait. If weeks and you have the ability to be flexible, then don't wait.
I am lucky though to live in a borough that does things properly, doesn't have staff who give out wrong and unhelpful information working in admissions and seemingly haven't lost an appeal in the borough (for Reception application) for many years. Always been crystal clear to me.
Bear in mind the admission folks will be very busy right now with Primary admissions, it should quieten up again in a month.
Also in September when schools open again. V busy.
And don't try to get information through the summer holidays when schools are closed, the council won't know the answers. For example, one of my DD classmates didn't come back at all in Sept. The place was filled two weeks after the start of term straight from the waiting list.