Some of this is laid down by the government.
The authority (Surrey CC) MUST have co-ordinated admissions for the normal year of entry (i.e. reception).
You will be applying outside the normal admissions round. The authority MUST co-ordinate these admissions too, although some do insist that parents contact their preferred school first.
Assuming your children are British citizens, you are entitled to apply for a place NOW. Don't let them try to put you off. If they tell you to wait until you are in this country, point them at paragraph 1.61 of the School Admissions Code. The code has the force of law so they MUST follow it. However, although you can apply, they don't have to allocate you a place as they cannot be certain when you will be in the UK or where you will live when you get here. Plans can change right up to the last minute.
There are specific provisions in the Admissions Code for applications outside the normal admissions round. In particular, they MUST handle late applications in line with their co-ordinated admission arrangements and they cannot refuse a place simply because you have applied late (paragraph 3.24 in case you need it). This applies regardless of the reason for the late application.
I would suggest that you get an application in for your DS as soon as possible. You don't have to wait until the appeals are finished or, indeed, for anything else. If Surrey CC try to make you wait, kick up a stink and point them at both the paragraphs I've mentioned above. Surrey CC may well want some specific evidence to prove that you are moving into the area - make sure you supply this as soon as possible. Before applying, make sure you understand the admissions criteria for your preferred schools as they will govern how likely you are to get in. Hopefully this will get your DS sorted out without any problems.
Your DD is another matter. The problem you may face here is that there is a statutory limit of 30 for infant class sizes. If the school already has that many in the class your DD would join, they cannot take any more. However, the good news is that there are some exceptions. As you are moving into the area, your DD may be classed as an "excepted child" (which means they have to admit her even if it pushes the class over the limit) if there is no other available school within a reasonable distance (paragraph 2.63(a)). Unfortunately the regulations don't specify what is a reasonable distance but this would give you a good case for an appeal if, say, they allocated a place at a school 10 miles away.
I sincerely hope this turns out to be straightforward. If not, contact me and I'll do my best to help.