What controls do state actors in Britain have over otherwise banal tech orders from China, in order to trace them post hoc?

Depends how much they cared, I'd wager. GCHQ operate a "full take" surveillance apparatus, so the data is certainly there.

Honestly, I assumed you'd order the parts from a UK retailer.

I think you could quite easily make something out of generic parts. The traditional material for remote control planes is tissue paper and balsa wood. The controller could almost certainly be made with a generic micro-controller, and battery cells are more or less the same whether they come from a phone or a drone (afaik).