The burser at the school should be the one chasing your payment for you that’s who I’ve dealt with in the past when I’ve done them,although the majority of this stuff is done for half what it used to be cleaned for.
I looked at a school around 5 years ago and it was priced conservatively at around 3k,someone travelled down from north London and cleaned it for 1200 with 2 other workers,the caretaker actually said to me when quoting they do a terrible job they want rid I said I’m not surprised at that kind of price. Spoke to him 6 months later and said I didn’t get it then,he said nah they kept the same company on lol,waiting for payment for up to 6 months is not unusual for schools in my experience. Schools and the local council ruined a roofer I know because of late payments,he was owed in the region of 200k the bank waited and waited again but in the end even when he took all his paperwork showing it was owed from the government they still pulled the plug on him.