You can apply now, but the problem will be that they won't hold a place for him; if you get something now, you will be expected to take it up within a couple of weeks.
Also, not all Surrey schools have their in-year admissions handled by surrey. Check on the schools directory
www.surreycc.gov.uk/schools-and-learning/schools/directory-of-surrey-schools
whether the ones you are interest in do. If they don't, you need to apply direct to the school itself.
So for your eldest, you can't really do much until the latter half of the summer term, when you could make a case for starting in Sept as the start of the next half term. Proof of exchange should be enough.
For your reception child, there are a couple of options. You could make a late application to your preferred school once you have exchanged, or you could wait to see which school your eldest gets (which won't necessarily be your nearest, it could be some distance away) and then hope to get your youngest in to that one then. Which way round you choose is up to you, and will probably depend on your ability to transport the two of them and possibly defer the youngest for a term or two whilst you wait on waiting lists for an acceptable space.
However, the chances of them ending up at the same school, at least to start with, are slim; surrey schools are oversubscribed generally, so you will be put where there are spaces - your reception application will miss the normal allocations round so places will have already have been given out, so both boys will be given spaces wherever Surrey can find them, which could be a couple of miles away at different schools I'm afraid. Having a sibling will bump you up on waiting lists, but it won't get you a space if there isn't one.