Child passport renewal appointments?

Hi everyone. Been a few years... All settled with ILR with thanks to support on here along the way. I now have two child US passports that need renewing in person--one expired during lockdown and the other due to expire in December. The embassy in London is booking appointments again but they seem to be instantly filled at the start of each week, and they are not booking ahead. Has anyone figured out when the appointments are released? Thanks!

P.S. I realise this probably belongs on a different page/topic--but couldn't locate the right one!