you should be able to find a nursery place although you may find it is in a proper nursery rather than a pre-school
in terms of you getting ds2 into reception, this will depend on where your house is - hopefully the LEA have told you which schools you are most likely to get a place at
with respect to your ds1, I hate to tell you this, but that school year was the most oversubscribed they have ever had in Teddington to date, even though Collis now officially has 3 classes a year. There is a woman who lives on Bolton Gardens (she moved there after the application date) who did not get in and she is still on the waiting list for reception (this year, so yr 1 next year) and is hoping that once her second child gets a place in the school, her first one will also get sibling priority.
If you can afford private schools, it may be worth exploring that option, even if you use it as a half way house. The ones most commonly used in Teddington are Newland House school , The Mall School and Twickenham Prep . The Mall is boys only I think, the other two are mixed. We have heard lots of positive things about Twickenham Prep - I think the head teacher is very well respected (and has the sort of all rounded approach to schooling - the 'every child will thrive' rather than pushy ethos).
Also, if you do eventually manage to get a place at a state school, do not worry about them joining at funny times in the year. There are loads of kids in Teddington whose parents are in the oil industry or in international jobs who scoot off for a year or so (on foreign postings) and come back in. The head teachers and children seem to handle and manage this situation very well!
Good luck, let us know if you need any more info.