Hmmm, Nottingham.
I think I can say with some degree of certainty at this point that Nottingham did indeed have proper kiosk type Police Boxes from a fairly early date.
In just looking at the OS Maps, with a rather cursory and random search, I was able to find at least nine PCB/Box marked objects and one PTP/Post, all dating from 1954 or earlier (no OS Maps of the right scale seem to be available for the area from the 1930s and 1940s), with most continuing on into at least the late 1960s. Interestingly, the Old Market Square Box that started this thread only shows up in the 1960s and later OS Maps, so it appears to have been a late arrival.
In looking at other sources, it would appear that the original Nottingham Boxes were shed types which were installed as early as the mid-1920s, after Crawley's success with the concept in Sunderland and Newcastle. In fact, Nottingham was one of the cites with installed Police Box Systems that Gilbert Mackenzie Trench and Assistant Commissioner George Abiss visited in early 1929, when they went on a fact finding mission while finalizing the Met's own kiosk design and Box System prior to the final experimental installations at Richmond and Wood Green. They also visited Salford, Manchester, Sheffield, and Birmingham.
I don't have the specifics of the Nottingham system when they visited it at hand - I'll have to look up my copy of their report. According to John Bunker, however, by the late 1920s Nottingham had "a number of timber built boxes (7 feet 6 inches by 5 feet 8 inches) operating with blue signal lights."
Nottingham apparently also had some 100 "call points" on the streets (probably of the pull alarm type) which were connected to both the Fire Brigade and the City Police, who were thinking of installing their own proprietary Pillars (with telephones) to replace these as late as 1937, despite the GPO's own PA1 Pillars then being in common use throughout most of the provincial Police Forces by that time.