Bee Math has a queen emerging 16 days after the egg was laid or 8 days after being capped. I think what you're referring to is that it takes about a month to get from egg to *laying* queen (hatched, hardened, mated & settled).
The drone population should be peaking just about now, so there should be plenty of eligible mates in any nearby DCAs. Around here, the workers generally don't kick them out until September/October.
I have a couple of mating nucs where the queens never returned, so I gave them each a frame of mixed brood yesterday so they can start all over again. Last year, I had an emergency queen develop & mate near the middle of September. She did just fine.
Still plenty of time yet. Just throw the feed to them when you're growing a queen.