Pashley Pickle child's tricycle wheel rims to buy?
I edited the question so it wouldn't be a strictly sourcing/shopping question.

It appears these use the 16" size with bead seat diameter 349. Look carefully around the tires and old rims to try and find the number 349 to confirm. It will likely be on the tire and say something like "38-349". If you don't find it and find some other 3-digit number instead, that's also your answer.

Sometimes there's no such marking. If that's the case you can identify what BSD you're dealing with via measurement.

349 is the Brompton size. Rims for it are readily available but are a niche product you need to get from Brompton shops/dealers/etc. You also need to match spoke counts. Many or most Bromptons use 28h and it appears this trike also does.