I'm going to answer my own question, as it was not a straight-forward process, but it all worked safely in the end.

1. After downloading the Mountain Lion installer from the App Store, I ran it, but it wouldn't allow me to install onto my "Macintosh HD", giving the message "This disk cannot be used to start up your computer".




2. The solution to this is to resize your target install permission to provide 128MB of space afterwards, as described here. However, my disk couldn't be resized as disk utility produced the error "MediaKit reports no such partition".




3. However, I was able to run disktutil resizeVolume from the term, but this failed during the validation stage due to errors in the filesystem. Disk Utility was unable to fix these online, so I needed to boot from an OSX install disk to run Disk Utility from there.




4. At this point I ran into an issue that my Leopard and Snow Leopard install CDs wouldn't boot up correctly (the former just hung, the latter kept restarting my system). Luckily I had a Lion install CD, and this was able to boot and I could run Disk Utility from there to repair the partition. Interestingly, I couldn't resize the volume from there (it gave an error that the filesystem type wasn't correct, even though it was), so




5. I had to boot back into Snow Leopard and run the resize from the command line.

At this point the installer ran and let me select the newly resized partition. The partitioning for my dual boot setup meant that the installer warned me that "Some features of Mac OS X are not supported for the disk Macintosh HD", but that as OK and the installer ran fine.

After this Mountain Lion was installed fine but my rEFIt boot menu had vanished. This was an expected part of the update process (as documented here) and I just had to run

cd /efi/refit

./enable.sh

and the menu was back at the next restart.