Did anyone else notice the doors of the genuine Police Box in the first scene of "An Adventure in Space and Time"? When the policeman came out, the doors opened INWARDS, like those of the TARDIS-- not outwards, like those of a proper police box. "Sexy" would not be happy!
The answer is obvious. The police officer is the real Doctor who wanted to meet the man who starred in the series he inspired. The Doctor just couldn't let on that he was a real time traveling alien and not a policeman.
Well, it's not like each bobby had his own police box. They were regional stations where officers could contact the precinct, get supplies, confine people until transport arrived, etc., or where civilians could summon police help by using the telephone and activating the flashing beacon.
Given how accurate they tried to be with everything else, I was surprised at some of the inaccuracies with the TARDIS prop itself. Like the missing frame around the door sign, the extra bar under the windows being painted blue instead of white, or the base which wasn't close to the right shape and which was also painted black... for some reason. Given Gatiss's love of the era, I figured if anything was going to be as accurate as possible, it would be the TARDIS.
A lot of things in the movie weren't accurate. Like Daleks saying "Exterminate" in "The Daleks," which is an anachronism. Or a lot of the chronological details of the production, like when the first Annual was published, when Lambert left, etc. The movie was more an impressionistic distillation of the early history of the series than an exact recreation of it.