Following the advice on this forum, I have overcome my unreasonable fear of making phone calls to other countries, and finally got around to ringing the Firenze museums booking service a few minutes ago!
I am happy to report that it was a painless experience and that I got my second choice of date, (For your information, Tuesday 1st August is almost fully booked for the Uffizi. The operator told me that Tuesdays are always the most crowded.)
As we will have two kids in tow, we won't be spending hours and hours in the Uffizi, and the helpful phone operator said that most people find an hour in the Accademia plenty of time. I've therefore booked both visits for the same day, beginning with the Uffizi in the late morning, as we are staying an hour's drive from Florence and the Accademia at 4pm.
The good thing about booking directly with the museum service is that if you don't make it to the museums on the day you've booked, (always a possibility for us, due to oversleeping, forgetting, getting the day wrong, etc.!) all that happens is that you miss your slot. You don't have to pay anything until you arrive.
Only a few more days to go before we arrive in Casole D'Elsa - I can't wait!!