"Do a shop" for "go shopping"

If you don't use this phrase, you may not know that it carries some precision. If I say that after I do X, I think I will have time to go shopping, you don't know if I just intend to randomly browse, get birthday presents, or buy the weekly groceries. If I say I think I will have time to shop, then the random browsing is excluded (among the people I talk to) by that phrasing. And if I say I have time to do a shop, it's marked as a sort of to-do list item of regular occurence, meaning the weekly "grocery shop" that has to be done at a regular rhythm. The reason we have three ways of saying the same thing is because they aren't precisely the same thing.

If the noun a shop is taken to mean the act of shopping then do a shop is the same grammatical construction as phrases like do a backflip. I don't know where it came from but it is certainly common idiomatic usage here in the UK. As @BarrieEngland states it is most often used with an adjective (big, weekly, etc.). It is generally premeditated grocery shopping - not just picking up a couple of things and certainly not shopping as a leisure activity.

The OED’s definition 2e for shop is ‘An act of shopping for purchases’ and it is described as colloquial. The earliest citation in this sense is from 1960: You should find it possible to have one big ‘shop’ a week with a small mid-week ‘shop’ for perishables. Once we see shop used as a noun in this way, any grammatical difficulty over it following do disappears. It's fairly common in the UK, but shop is more likely to be preceded by an adjective such as big or weekly than not.