Bob, this song is known in New Zealand too, from the 1920s era.
I recently talked to a lady (aged 92) whose father wrote a trade union parody of 'Only a Leaf' around this time and she remembered the song.

It could be that T.B. Fayme was a British composer. There are three separate sheet-music songs by Fayme in the British Library catalogue: Only a leaf, His Charity Covers his Sin & Ride-a-cock Horse. These are dated between 1898-1905 and all published in London (apart from one which is a pirate edition). To check these click:

Here

'Grandad's Songs' I believe is mid-1930s - my copy has a 1934 copyright notice.