I fixed this myself. I suspected that the problem was actually a Windows-like issue, and I was right.
Just like Windows will freeze when you click the Save In dropdown list while it spins up the CD/DVD-ROM, CXOffice was freezing because the CD-ROM, floppy and a handful of other mounts were mapped to various drive letters. However, it couldn't access these properly, so was simply freezing and then (I presume) timing out after some time.
The solution was to go into the CXOffice folder and remove the shortcuts. On my computer, I found the shortcuts at ~/.cxoffice/win2000/dosdevices/. I removed all of them except C: (the Wine fake C:\ drive), Y: (mapped to my home directory), and Z: (mapped to root of the Linux file system). After this was done, Save As appears instantly, and clicking the Save In dropdown list responds instantly too.
Incidentally, I forgot to congratulate the Wine developers. Wine and CXOffice have come on in leaps and bounds recently. I remember trying CXOffice a year or two ago and it was very unsatisfactory. There was a lot of graphical glitches and even the occasional crash. But this new version lets Office run like it's on a native Windows system. Well done, Wine developers! You've produced a ten-ton rabit from a hat and made it appear effortless. 😊