The real historic hostility between Tottenham and Woolwich goes back to the circumstances in which Woolwich were elevated to the First Division in 1919/20, the first season after the resumption of league football after World War 1. Relations would have soured between Spurs and Liverpool and Man U in particular as well and Chelsea to a lesser extent arising from all that business as well. There was essentially a big conspiracy with anti-semtic overtones between those four clubs and others to get Spurs out of the top flight and Woolwich in.
It was agreed, shortly before the resumption of the league after World War 1 in 1919 that the league was to increase from the 20 sides from the 1914/15 season immediately prior to World War 1, to 22 for season 1919/20. The precedent on any prior increase in numbers from the inception of the league in 1888/89 was that there was no relegation. On the previous increase from 18 teams to 20 teams for the 1905/06 season (in which Liverpool were promoted), there was no relegation of the bottom two sides from the 1904/05 season.
It was initially agreed that Chelsea and Spurs who finished in the bottom two relegation places in 1914/15 would remain in the expanded 22 team league for 1919/20 and the top two sides from Division 2 in 1914/15, Preston North End and Derby were promoted. There was dirty tricks going back to 1914/15 with Man U also in the mix to go down that season and ultimately finishing third bottom and out of the relegation places by the skin of their teeth. Itāll probably come as a great shock to Man U supporting Liverpool haters like @BruidheanChaorthainn to hear that Man U finished just ahead of Chelsea and Spurs in 1914/15 only by conspiring with Liverpool to fix a 2-0 win for Man U when they met on Good Friday, which ultimately ensured Man Uās survival.
Just prior to the start of the 1919/20 season, there was a late u-turn and it was decided that Chelsea and Tottenham both had to be re-elected to the top flight. Chelsea were ushered back in no problem. When it came to the issue of re-election for Spurs, it was decided by the league clubs that it would go to a vote. Instead of a straight vote between Spurs and the 3rd place team in Division 2 from 1914/15 (Barnsley), there were 7 teams put up for election for the 22nd and final slot in the top flight for season 1919/20. Woolwich who finished 5th in Division 2 in season 1914/15 secured election to the top flight largely on the back of a pact between the Liverpool owner, John McKenna, Man U owner John Henry Davies and the Woolwich owner Henry Morris.
Spurs went and won Division 2 in 1919/20 to get promoted back to the top flight for season 1920/21.
I believe Woolwich in 1919 are the only side in the 132 year history of the football league to get promoted to the top flight in the boardroom and not on the field of play.