Just to round this out:

The landlord ended up replacing the whole toilet - apart from the stain it was also leaking, with water from the cistern streaming into the bowl every time it was flushed.

I did ask the plumber who did the job about the staining on the old one, he said it wasn't actually limescale, and it would possibly have been a manufacturing defect as he had seen this before. It was the original toilet that came with the property, so it was well over 20 years old.

In any case, I have a nice new toilet now, and no more water leaking into the bowl after flushing 🙂