Not exactly, I'm afraid, Joanne. However, the abbreviation 'bike' itself dates only to the 1880s and 'the local' - ie 'the neighbourhood pub' - is from the 1930s. My guess is that 'local bike' would definitely be post-war at the earliest, just from its general tone. I have also heard the version, 'the college bike', which suggests that the whole wretched idea might have been an invention of male university students which was then taken over by the public at large.
My reference to the pub above, by the way, in no way suggested that the 'local bike' was a loose lady connected just with the pub! No...I meant only to illustrate that such uses of the word 'local' itself were relatively recent.