OLD GAMES
I would recommend Dan Gindlings "Bicycle Collectibles with price guide", p 57-64 (the book is much better than what its title suggests) for a history of anglo-saxon ninteenth century cycling games. He mentions the following games.
Junior Bicycle Game (Parker Brothers, 1897)The Bicycle Race and Skating Rink (Milton Bradley, 1885, a five-in-one game)
Bicycle Wettfahren (Verlag von Moritz Ruhl, Leipzig, 1890) (according to Pryor Dodge's book
"The Bicycle", it is from 1887, and is called "Das Bicycle Wett-Fahren")
A Cycling Tour (J.W. Spear & Sons, London, ca 1890)
Game Of Bicycle Race (McLoughlin Brothers, 1891)
The Bicycle Race - A Game For The Wheelmen (McLoughlin Brothers, 1891)
Bicycle Race (J.W. Spear & Sons, London, late 1893)
The New Bicycle Game (Parker Brothers, 1894)
Wheeling (J. Jacques & Son, London, 1896)
The Century Run Bicycle Game (Parker Brothers, 1897)
And some cycling-related games:
Game Of The Telegraph Boy (McLoughlin Brothers, 1888)
Game Of The District Messenger Boy (McLoughlin Brothers, 1904)
The Messenger Boy & Checkers (Milton Bradley, 1930s)
To these I would add at least (though it's sometimes hard to say if they are the same game):
The Boston Sunday Globe game (published 1895, August 4th)
The Bycicle Game (Milton Bradley, ca. 1895)
Bicycle Racing (McLoughlin Brothers, 1900)
Jeu du Vélodrome (NK Atlas, Paris, 1900?)
Bicycle Race (Milton Bradley, 1910)
The Popular Game Of Biking (Globe Series of Games)
Le Jeu des Bicyclistes (Pierre Patau)
And maybe even some later games:
Jeu des courses cyclistes derriere entraineurs Montlhéry Record (Edwill, 1925)
6 Day Bike Race (Lindstrom Tool & Toy, Co, 1930s) not exactly a board game
Jeu du Veldiv (H.B. Editions, 1933)
The Bike Race Game (Master Toy Co, 1940s)
I am sure there must be more games, specially European. Additions welcome.