Theres one down here in the depths of Cornwall. Havent been there but keep meaning to. Couldnt find a web site but found the following description on tinterweb
The British Cycling Museum has more than four hundred examples of different cycles in addition to an old cycle repair workshop, with tools from long ago. There is an extensive library of books here and such interesting articles as the first cycle oil lamp. As well as lots of old medals and cycling memorabilia, there are window displays of gas, candle, battery and oil lighting. A gallery contains framed cycling pictures, ceramic cycling items and models of cycles. The museum documents cycling history since 1881.
The museum is housed in an old Victorian railway station and has been open since 1992. It is open from Sunday to Thursday each week throughout the year and is situated a mile north of Camelford on the Boscastle road.