A nut size is determined by the size of the threaded hole. Thus, a
1/4-20 nut has a nominal 1/4" hole with 20 threads per inch. Metric nuts, same story, but pitch/size is expressed differently.
The wrench and/or socket you use bears no clear relationship, obviously. The size of the wrench you use is whatever fits snugly, and is expressed as the size across the flats.
I think the Witworth (semi-obsolete UK system) measures across the points. I remember on an old Austin-Healy (gee, how I miss that car!) that I had problems with my US wrenches fitting properly.
For US nuts/bolts, use fractional inch wrenches and/or sockets. For metric ones, use Metric wrenches/sockets. Some sizes are pretty close. I notice that the last set of 3/8" drive sockets I picked up had both fractional inch and metric sockets in the set.
You need both, as I've found that my 2000 Jeep Cherokee has both metric and US fittings!