I have had a drive fail in my RAID(1) array on my home server and I'm wanting to get the system back up temporarily while I obtain a replacement.

Unfortunately two of my RAID partitions have not come back. I have done an "mdadm --examine --scan" and this now shows something like this:
Partitions /dev/md2 and /dev/md5 are not mounted and appear to have been renamed /dev/md/2 and /dev/md/5.
The underlying partitions were/are simply /dev/sda2, /dev/sdb2 etc.

Is there some way of forcing the /dev/md2 array to be created on boot or do I need to create a new array using the updated "/dev/md/2" terminology ?
(Note: I'm OK that this runs degraded in the short term.)

Has the UUID changed? How is the array referenced in /etc/fstab?

No the UUID is unchanged. I just unplugged the failed drive and rebooted.

I tried adjusting the names in both /etc/mdadm.conf and /etc/fstab to refer to /dev/md/2 but this made no difference.

Have a look in the initramfs.
Easy to test with a liveCD rather than the system itself - assemble the array(s) yourself and mount them anywhere just to check.