I could plug the drives into my windows pc? See if it recognises a volume?
A Windows pc is limited. But it can see if the disk is recognized at all. If the disk doesn't show up at all, a Linux pc won't see it either.
Further it can see if the partition table is correct. There should be 2 partitions, one of about 500MB, and one containing the rest of the disk. *Do not* agree when Windows helpfully offers to 'initialize' the disk.
Some software exists which might be able to read the disks on a Windows PC. UFS Explorer can be helpful, although your disk probably has a ReiserFS filesystem, which is not supported, AFAIK.
Further a program like PhotoRec might be able to rescue your files, without any help of the filesystem.
But. If the disk is already dying, intensive surface scans might push it over the rim. The recommended way is to create a bit-by-bit copy on a new disk, and do the recovery from there. On a Linux system it's no problem to dump the disks in two files (assuming the disks are both readable) and do recovery on that files.
I have a horrible feeling that the volume will be gone and I will end up running one of those file recovery programs that just manages to recover files by examining bytes and looking for data that looks like known file types
Possible. And why is that horrible? It's horrible that you have valuable data without proper backup. It's great that software exists which can recover files without any help of the (possibly damaged) filesystem.