The information here is that which I have so far managed to glean from various sources, but cannot be guaranteed as correct. If you have other information relating to the history of cycle paths, please contact John Franklin.
1890s: First cycle tracks in Netherlands. Reference in Dutch "Bicycle Master Plan" of mid 1990s.
1895: Ocean Parkway bike path opened in Brooklyn, New York. Said to be the oldest path still in existence today. Other early bike paths believed to have been constructed in USA, especially on the east coast, but construction ceased at turn of century as cycling waned as a common mode of transport. It seems quite possible that the US invented the bike path.
1897: First German cycle tracks, in Bremen: 2.5m wide in centre of road. Main motive was comfort of cycling in days when most roads were cobbled or rutted.
1898: Cycle tracks implemented in Magdeburg. 400km by 1929. Most financed by Studiengesellschaft für Automobilstraßenbau.
1899: DEM250,000 spent on cycle tracks in Hamburg.
1900: Cycle tracks introduced in Lübek. Paid for from receipts from number plates that had to be purchased by all cyclists. Tax removed in 1919, but obligation to carry number plate remained.
1901: Demands for cycle tracks in Germany to foster cycle tourism.
1910: German cycle tracks moved from centre to sides of roads.
1920: Quote from first Dutch Roads Congress: "After all, the construction of bicycle paths along the larger roads relieves traffic along these roads of an extremely bothersome element: the cyclist."
1920s: Mass construction of cycle tracks in Germany. Motive: to remove disturbances in the fast flow of motor vehicles caused by cyclists. Propaganda cited paths as pro-cyclist, and first use made of 'safety' argument to get cyclists to use them. Many arguments between police and cyclists, the latter prefering to use the newly tarmaced roads.
1926: Cycle tracks made compulsory for cyclists in Germany.
1930: First tarmac cycle tracks in Bremen. Some private paths created, to use which cyclists had to pay an annual fee.
1930-1 Programme of cycle track construction by ANWB in Netherlands (by this time as much a motoring organisation as a cycling one).
1930s: Main construction of cycle tracks in Denmark, France and Switzerland. Earlier Danish and Belgian tracks were said to be much better quality than those in Germany.
1930s: Cycle track construction in various places in UK. (Track on London's Western Avenue, built 1934, believed to be oldest existing UK cycle track)
1934: New German legal instruments to address "the problem of disciplining cyclists" who did not use cycle tracks. Bicycle associations outlawed by Nazi regime. (Source notes that by this time the legal obligation to use cycle paths already existed in most countries.)
1936: All new roads in Hannover built with cycle tracks. 120km by 1936.
WWII: Use of cycle tracks made compulsory in Netherlands, under Nazi occupation.
1967: Bike lanes installed in Davis, California. First known reference to a lane as distinct from a track.
It seems that the first cycle paths were to meet the needs of cyclists in terms of comfort and ease of riding. Soon after road standards improved, however, the motive for building tracks changed to one of getting cyclists out of the way of motor traffic. Only in the Netherlands does there seem to have remained a pro-cycling reason for building paths.
There is little evidence of cyclists demanding cycle paths for reasons of safety until the 1970s; indeed much of the opposition to using paths in earlier years was on account of the added danger present.
From Cycling Lanes to Compulsory Bike Path: Bicycle Path Construction in Germany, 1897 - 1940, Volker Briese, The 5th International Cycle History Conference, Cambridge, 1994. ISBN 0-933201-72-9
Radwegebau, Dr Ing. Schacht, c. 1938.
Cycling Digest Infrastructure Index