Okay, I've figured it out. I have to say, this was with no help from Adobe's customer support who spent an hour on my machine, mucking about with things, making claims that are just not true before I eventually had to call them out for not knowing what on earth they were talking about and kicking them off.
The solution in the end was that one of the songs I'd used in the slideshow wasn't working. It was good enough to play normally as a track, but presumably had sufficiently corrupted metadata or some other attribute that was crashing the Slideshow module if it was used anywhere in the track. Deleting the song and replacing it with another has fixed the issue and everything is now working as it should.
Things tried or claimed by the Adobe customer support human (all without success) included:
- just pressing play 10 times in a row expecting something different this time
- delete all your preferences you've spent 9 years getting how you want
- making LrC run in Rosetta emulation
- Slideshow only works with Collections (absolute rubbish)
- Slideshow won't work if it contains any photos that have been synced
It was these last two (blatantly wrong) claims that convinced by the guy had no idea what he was talking about so I kicked him off before he tried to delete all my syncing data.
But, anyway... final answer in my specific scenario was a corrupt music file in case anyone has a similar issue in the future and want something else to try!