So far I have 23 fully made. They're supposed to be small/newborn size, but
they seem to look fairly big to me, so I'm pretty sure they'll last a while.
I do have a washer and dryer, but I don't care to do tiny loads of diapers
daily... I'd rather use a diaper pail and do a load of diapers every 3 or so
days...
Any suggestions on how many diapers I should have handy and ready to go?

Sounds like you have plenty, the ideal situation is to have enough such
that you can go between relatively full loads in your washer. We used
flat terry squares with our first, we intially had 24, but 12 of them
were cheaper and they weren't so good (I don't actually remember why!),
so we bought another 12 of the first brand, that seemed to fit our
washer well, I guess 15-18 per load and a few to use whilst they went
through. I think an American style washer would probably take more, but
then I think we were already at about my limit for how long to store
them (2-3 days).

Of course if you have other children in diapers, as you do, the amount
per child per load goes down and we found we didn't need anymore diapers
when #2 arrived, though I was glad we had the extra dozen bad ones,
because occasionally we would get to the last of the good ones.

We moved to prefolds and never owned more than 18 in any size, I think
they are slightly bigger overall that flat terries, so nappy bucket and
machine took less.

I suspect with 3 kids in diapers you're going to be at the almost
needing to do it daily stage, which cuts down how many you need!

Cheers
Anne