If your DS is currently in Year 6, if you haven't already submitted a Common Application Form (CAF), you can submit a late application now. It must be done through your current home LA, but you can name state schools anywhere on it (e.g. if you live in Hertfordshire now, you use the Herts Form, but nane schools in the Twickenham area as highest preferences).
At this stage, the application will not be considered until after national offer day (3rd March this year, as the 1st is a Saturday). As your home address won't have changed by then, your home LA will initially allocate a place at its nearest school to your home address which still has spaces after all on-time applications have been accommodated.
You might be lucky, however, and find that one of the new area schools you have applied for didn't fill all its places with on-time applicants and late applicants ranked higher against the school's oversubscription criteria than your DS; in this case, a place at that school will be offered.
In any case, your DS will be / you can ask for him to be (check specific rules) placed on the waiting list for each school you prefer to the one allocated.
As for your younger DC, unless you are applying for a year 3 place at a stand-alone junior school (where year 3 is the normal poinyof entry), you will be making an in-year application. If you do that now, for year 2, if there is a place, your DC will be expected to start 'now' (the place won't be held over for a September start); are you in a position to do that? If not, you will have to wait until much nearer to the time of entry to apply.
If you are applying for a September start at a Junior (not primary) school, then you have also missed the deadline for an on-time application. You need to look at both your current LA's and the relevant new LA's requirements for applications for a place for thos DC.