Thanks for all these further comments and suggestions! (Sorry I've been a bit late replying, but I've been away from the computer for a few days.)
Nick29: yes, I do think he had to 'live on his wits'! To clarify: the first record I have of him is 1900 in London joining a Irish/Canadian regiment - he called himself James Aloysius Smith, birthdate 1881, Ottawa. In 1905, he deserted the army and joined the Royal Navy (still calling himself James Aloysius Smith, birthdate 1883, Ottawa). A year later, while still in the navy, he married a Lambeth woman (his first wife) - he called himself John Arthur Shaw on his marriage certificate. Soon after his marriage, he deserted the navy and set up home in Lambeth, where he worked as a gas stoker. In 1910, there was a King's Pardon regarding army deserters and, for whatever reason, he rejoined his Leinster regiment. He had to assume his original 'army' name of James Aloysius Smith, so at this point, he's using two names!
After WW1, he returned to England and continued to live here for the rest of his life as John Arthur Shaw (he died in 1975, aged about 95). I have a Canadian passport issued to him during the '50s - I have gathered via the Canadian Archives site that records of old passport applications no longer exist, and also that anyone applying for a Canadian passport around this time didn't have to provide any verification of nationality. Apparently, my grandfather was always very cagey about his past, though he told his children that he was an orphan, was originally from Ireland (Tipperary is his birthplace on his obituary), and that he was brought up by the White Fathers, or some such religious order, in an orphanage in Canada. I've tried to track down orphanage records in Ottawa, but drawn a blank.
As you can imagine, it's really difficult to unpick all this: were Smith or Shaw his real names? was he born in Canada or in Ireland (someone has looked up Ottawa births for the various names in 1881 and 1883 and not found him)? It's almost as though he was constantly running away from his past, and throwing up smokescreens along the way!
Thanks, Redroger and jds1949 for your comments... as to the name 'Aloysius', yes, I have wondered about the link to Catholicism, though he put on his army records that he was Protestant; his second wife (my grandmother) was actually a Catholic, but I don't think he ever went to church with her!
Andcarred: my uncle (the only one surviving of my grandfather's 5 children) also has Dupuytren's, and has had to have some of his fingers amputated. I met him for the first time a few years ago (he lives in N Zealand) and the first thing he did was examine my hands!
Many thanks again, everyone - all food for thought for a desperate woman!