Virtually anywhere the public have authorised access to is considered "public" for all sorts of things, in the case of motoring law that means that a supermarket car park during business hours is "public", a normally "private" garden or field during an event attended by the public is "public" for the duration of the event, a drive through is no different to the supermarket carpark in those regards
Basically if it's not gated and you don't need special permission to be allowed to drive there it's likely to be considered "public" for the purposes of the RTA. A Tesco car part during open hours is likely to default to being a "public place", that same carpark at 1am in the morning when the store is closed and a gate has been closed preventing public access to it is likely to be "private".
This has been tried and tested in all sorts of cases about all sorts of parts of the RTA, in loads of different circumstances and locations over the last 50 or so years.
It's much the same as you can be in a "public" area in regards to expectations of privacy, even if it's private property and you're inside a building, the definition isn't about ownership, but reasonable and common access.