..... and the winner is:
Sheffield F.C.
...... nothing to do with Sheffield Wednesday or Sheffield United - a completely different club, still going and currently playing in the Northen Premier League, Division 1 South.
Sheffield F.C. is the world's oldest football club, founded in 1857 (156 years ago) and have never played in the top flight, since the league started in 1888.
I suppose technically speaking, no team could have been waiting more than 125 years, 'cos that's when the football league first started.