It grew in a small pot of peat free compost in which I had a small yew tree seedling. At first I thought it was a fern as I often find ferns of various kinds growing in pots indoors and out.
I separated the plant from the yew and shortly after took the first photo, which includes a 50p piece for scale. The second image shows the leaves in more detail. I have since re-potted the plant twice and the third photo shows the present appearance, including the white threadlike roots.
The last image is from a scanner which I set at high resolution to show part of a single leaf.
I have searched the internet (probably asking the wrong questions) and my sister has sought advice from the knowledgeable folk in her allotment association but we have had no success.
The plant seems to like being constantly moist and in strong sunlight (when it decides to shine).
I think that your plant is likely to be Ranunculus scleratus - Celery-leaved Crowfoot. Its a small flowered buttercup which grows in damp places - pond margins, river edges. etc. and does occasionally pop up in composts.