Okay - sorry but there was one big mistake in what Bestof7 said.
You only fill in one form even if you apply to schools in other boroughs/LEAs. You apply to the LA where you live, regardless of which schools you apply to.
Forget about the main application dates, you will not be here in time for those.
Your children will go to school in the school year with their age cohort. So if your 10 year old is 11 before/ or on 31st of August; they will go into year 6. The confusion is that you will be applying for two schools pretty much at the same time. One to go to immediately, and a secondary school for September. You will be too late to be considered in the "normal" application round.
However once the offers of school places are made in March you can join the waiting lists for school places at the schools you want. You will jump to the place on those lists you would be at if you had applied at the normal time. So if you live next door to a school you will be "probably" very close to the top; children who are fostered or siblings may be above you. As a lot of schools have movement on their lists you will have a good chance of getting a place.
When you get here the LA can advise you further.
Other things to know are, Catholic and C of E (and Jewish and Hindu) schools are part of the State system, but may use attendance at a particular church or other faith grounds as part of their selection criteria. So if you live next door you may be lower than someone who lives further away. There are some Grammar schools, selective secondary schools, in the London area, who select by ability. There are also schools which test all applicants, place them into broad ability bands, and the recruit a percentage of applicants from each band.
The LA is obliged to find you a school place within a reasonable time. If the place they find you is too far away, they have to provide transport, however in London this is usually in the form of the free travel pass. You children will be expected to get themselves to school by public buses (there are no yellow school buses). It is different somewhere like Guildford.
Your children are likely to have a gap between arriving and getting a school place. You can home school or give them a holiday.
BTW I don't know of a tubes stop or place called Holdon? Do you mean Holdon Road in Twickenham? You might want to start on the TFL website.