Thanks for the replies.
Fuzzyturns, the shed has a concrete base and dwarf brick walls. There is a timber floor on battens on top of the concrete slab. The dwarf walls are panelled on the inside with plywood. The floor has a damp-proof membrane which is carried up inside the dwarf walls and overlaps a DPC that sits between the dwarf walls and the stick-built top half of the shed so I don't think damp from below is an issue. At least I hope not!
I contacted a 'technical person' at Sadolin; they recommended cleaning with a fungicide (and since many are bleach-based, I'll probably just use bleach) then 2 coats each of a preservative and an exterior grade stain. He recommended waiting until the weather is warmer and drier to do the latter.
I think I'll also put in two wall vents to allow a cross-flow of ventilation.
Hopefully that will fix the problem