Where did ‘push-bike’ come from?

Books that one often used to see in secondhand bookshops, when there were such things, were the World’s Classics editions of the novels of Constance Holme. Humphrey Milford, publisher to the University of Oxford 1913-45, put all of them into the handy blue-bound format. The Spectator gave her second novel, The Lonely Plough (1914), a short but respectful review, declaring the book to be about Lancashire farmers, even though it repeatedly makes clear that the setting is Westmorland, the county where the author lived all her life.