I rebatteried the other UPS with three year old batteries. That went well, but when I restarted the attached servers:
One had a CMOS battery failure, and I had to replace the battery. This, of course, only manifests when you power the server off and on again.
Another had two drive failures. These are two 3TB Seagate drives, which are possibly the worst drives Seagate ever made. I replaced them with two 4TB drives, which I've found to be much more reliable.
The power distribution switch, that allows me to power on and off up to eight devices remotely, wouldn't switch on. I soon traced that to a blown fuse on the power plug. It was a 5 amp fuse, which is silly; it needed a 13 amp. That was quickly fixed.
So now everything is working fine, and the UPS will be good for another three years.