I did this a few years ago, at pretty much exactly the same time of year. i won't pretend it wasn't stressful...
Here's what we did:
DS was in Year 1. This meant that he was an application for a place outside the normal admissions round, and at least for us that meant the school controlled admission (in our area at that point, the Reception, Junior (if relevant) and Senior school admissions rounds were centrally managed but 'in year' admissions were managed by the school - in other words the head knew that he had a place in year 1, he knew DS1 wanted it, and he could allocate it to us and inform the LEA, regardless of where we lived).
DS joined Year 1 3 weeks before the end of Year 1.
DD was due to join Reception in the September. We applied when we knew we were going to move, but from our old address (40 miles away). We were a late application, so we were placed in the school nearest to the one we wanted with spaces as we were at the lowest of the low point in the admissions criteria (late, a long way away, no sibling). We requested a place on the waiting list. At this point we were something like 42nd.
We then moved to our new town. As soon as we had that new address, DD moved up the waiting list to 24th as we were no longer a long way away.
Then DS started at the school. At this point, DD became a sibling as well as living close, so she was 2nd on the waiting list (first was a girl whose brother started the same day as DS but who lived literally opposite the school).
A couple of people dropped out from those initially offered places...and DD was offered a place for the September 3 days before the end of the Summer term in July.
From my experience, I would say try to get your older child into the new school before the end of this school year, and place your younger child on the waiting list for reception. Once your older child starts, your rounger child should hop right up the waiting list due to sibling priority (in many schools, siblings get a place however far away they live so this may work even if you haven't moved yet!). If you can move as well, the closer distance shiould again move you up the list - and there you go.
However, if the school does not have a place for your older child in this current year, then i agree you have a dilemma as without that sibling priority it may be very hard to get a place for both.