TIme Machine IP Change Loses Backup History I have a time machine setup where the backup location is on a Synology NAS drive; having recently moved house, and ended up with a different router, I setup the NAS drive again, and it has a different IP address (formerly in the 10.* range, now in the 192.* range).

I couldn't - no matter what I tried - get Time Machine on the iMac to recognise the drive as the existing TM location; despite it showing up as an available disk in TM preferences. In the end, I deleted the old backups (nothing required from them yet, so it won't hurt to start again, I suppose). What I'd like to know, for future reference, is if/how I can get TM to recognise the NAS as the backup location if the IP address changes again. None of the hardware changed, so I can't see why this was an issue.
The new router (the hardware did change) is obviously using one of the default 192.* ranges instead of the 10.* range the old router was using. Your iMac and NAS picked up an address in the new IP range, (that's why the iMac could 'see' the NAS,) but because the IP in the NAT changed, Time Machine failed.

When you change routers in the future, you should make sure the iMac and NAS pick up an IP in the same range as the old router so there is no mismatch.