Many thanks, Mark.
This may have been my fault in the end. I have a very image-heavy site, and ticked the box to check image files as if they were executables. I strongly suspect that the previous scan didn't complete before the next scan was scheduled to start or something.
However, your mention of cron is interesting. In order to get another plugin to work - though, sadly, I failed - I wrote 'alternative cron' into the relevant WP php file. It could, perhaps, have something to do with that.
I can't do the check you've requested given the site had 13 hours' down time as it is courtesy of the host provider while I sorted it out and so, to be on the safe side, I've switched off automatic scans. If I switch 'em back on again to do the test you suggested, then if it does indeed all go haywire again, I'm being monitored and the host provider is likely to start jumping up and down on my head with greater intensity than previously.
For the moment, then, my sincere thanks for your help, and I'll mark this one up as 'resolved'. Lest anyone else stumble in here with the same issue, you may want to underline whether or not you think my scanning image files may indeed have been the issue in a follow-up post so others can see that.
My thanks, once again, for your help, Mark.