Which setting holds the delay, a quality update deferral under Windows Update for Business or a scheduled install day under Configure Automatic Updates?

Is your WUServer pointing to WSUS or going through Microsoft directly?

Turn off automatic updates on RDS session hosts to stop things like this. I'd patch two or three session hosts first and hold the rest for a few days. Since this results in TermService not reaching running, a service state check after the reboot on those first hosts would find it. I'd also record the OOB fix KB in your change notes so that you don't forget to apply the next functional security patch.

Disclosure: I build ET Ducky. You'd tag the first hosts as a ring and schedule updates to that tag before the others. The deployment pauses if services fail to come back after the reboot. https://etducky.com/documentation/patch-management