Bill Johnson was born on 11 January 1911 in Balmain, and died on 20 September 1974. His full name was William Jasper Johnson, and his nickname was Bunny (1 (http://nla.gov.au/nla.news-article239992149)).
↪ Something perhaps worth noting is that Bill Johnson's birth certificate says 11 Jan 1911, his baptism record says 10 Jan 1911, and his WWII militia record says 9 Jan 1911.
↪ One source reports that Bill was preparing for his 99th first grade game on 22 June 1940 (2 (http://nla.gov.au/nla.news-article236244280)), which appears to exclude the four matches listed for North Sydney in 1929. The following week for his 100th, another source says he was playing his 100th first-grade game "with the Balmain Club" or "with the locals [at Leichhardt Oval]" (3 (http://nla.gov.au/nla.obj-654387872), 4 (http://nla.gov.au/nla.obj-654391584)). I'm not sure if this 'with Balmain' phrase implies that he did in fact play games for another club. Source (3) also mentions he was "a product of the [Balmain] district," but then says he "first became associated with the Club in 1932."
↪ At this stage, I haven't found any source which proves or disproves that Bill played for North Sydney in 1929. I'm hoping someone might find something I have missed!
Bill Johnson's brother was Balmain's one hit wonder A Johnson (5 (http://nla.gov.au/nla.obj-637168218)), who was Alfie Johnson (6 (http://nla.gov.au/nla.news-article169364472)). He was born on 21 October 1917 in Balmain, and died on 25 April 1977. His full name was Alfred Herbert Johnson.
I have updated details of Bill & Alfie.
I'd be pretty certain that the W Johnson who played with Norths is not Bill as the Norths player was a hooker and Bill never played hooker also Alan Whiticker's Encyclopedia & Ian Collis' A to Z book both have them as separate players.
Also Norths at the time had a few Johnson's in the team Cecil, Harold & Jimmy were brothers W Johnson is possibly related to them, maybe you could dig into the 3 Johnson brothers and see if they had a younger brother who's name started with W.
I have also created W Johnson for Norths as a separate player to Bill from Balmain.