I'm back
Thanks for the replies all. My word, though, this gets complicated - by virtue of the fact that both my basins have funny-posh "pebble" plugs in them, and the pebble stays in there all the time. Then underneath that is a normal-ish thing that prevents anything massive going down the pipe.
The rods, I don't think I currently need to buy them - I'm sure the blockage is in the U-bend of the little cloakroom washbasin, everything else is running perfectly. So I'll get the equivalent of the Silverline drain snake that Andy linked to - the "Kitchen Craft and Sink Overflow Cleaning Brush" for a whole £3.25

which has brilliant reviews.
And one of the Knipex range, the individual item also has brilliant reviews, even if the drain snake does the job, it might be a good idea to have this little bit of kit.
I do have a plunger, but I don't have the ability to plunge *and* block the overflow, cos my arthritis is in lots of the necessary bits, like shoulders and fingers

but I appreciate that with normal physical abilities, it should work.
I'll let you guys know how it goes - thanks for the input all. I'm trying not to see the image of a blocked toilet in my head
