"United" isn't really a nickname per se, it's part of the club's name. Fewer than half of the Football League's many Uniteds actually formed from a merger--Manchester United, for example, was chosen by the former Newton Heath club when they decided to rename themselves, but they actually considered Manchester Celtic, among others. By and large, British teams named themselves "United" for the same reason that DC United did--because it sounded cool.
The actual nicknames--Reds, Gunners, Swans, etc. came about for the same reason that the original American baseball nicknames came about--because sportswriters needed to break up the monotony of referring to a club in the same manner every time.
Somewhere along the line, it became an American tradition to include the nickname as part of the team name, something which never caught on in British football.