Another possible origin of the ter 'The Old Bill'.

After ther Great War, a fair number of former soldiers either joined or returned to the Police.
During the War, Bruce Bairnsfather, the cartoonist had created the popular character 'Ole Bill', a sort of 'soldier's everyman'.
I believe that Bairnsfather was commissioned to produce some posters for police recruiting )or something similar), but even if not, it is a short step in association between the cartoon soldier and the ex-soldier in a different uniform - Perhaps some thought that the policemen truely had found the 'Better 'Ole'.

Walrus