Culprit identified....

[flick]

While weeding the back garden (this time with gloves on), I found another mystery plant, this time with white flowers, and Mike identified it as self-seeded nicotiana, the sap of which the internet confirms can cause skin irritation.

Fortunately, my diagnosis of the blisters as big but minor was correct: they've come off, and the skin underneath is undamaged. I'll have to remember that in future!