My plan is to stick my UniFi gateway / switch on a UPS, and get one of those power over Ethernet injectors / barrel jack extractors(?) to power the OR ONT.

Chatting to the BT guy that last visited, they reckoned the street cab would only last for half a day or so without power.

Yeah for mine I've got the unifi cloud fibre gateway but currently use its sole poe port for a u7 lite. Thats on an APC UPS which will give me 3-4hrs life. Then for the ONT I've just picked up a relatively cheap Eaton UPS that should give it approximately 3-5hrs. TBH any outage longer than that and I'm out of the house for the day until its fixed!