I don't know where you got your information from but it is almost complete bollocks.
From personal experience I know that it was sung quite generally some 20 years earlier, The original lyrics I learned were " ...we are the good King Billy's boys".
So far as I am able to ascertain it probably stems from late 19th century but I am still trying to tie that down.
I got my information from being brought up in the Calton, going to school in Brigton , travelling to games on the Mermaid Supporters bus during the 1960's and supporting Rangers for well over 60 years. I am very well acquainted as to the origins of this particular song and how it ended up as part of our supporters repertoire and I am talking about the lyrics as they've been sung since the 1920's. A few years back in another thread about the song I remember a poster on here referencing a newspaper report from the 1928 Scottish Cup Final and the journalist's description of a gang of men from Bridgeton singing it as they walked up Polmadie Road towards Hampden. If you don't know that TBB was the anthem of Billy Fullerton's gang from the 1920's then you are not well informed. If you want to take it back to the 19th century then the American Civil War anthem, 'Marching through Georgia' is where to start looking for the tune.