Hi, Barbara, many thanks for your message.
No, I'm afraid I haven't got any further with tracing my grandfather - where he went to after he and my grandmother separated is still a mystery (apparently my uncle thought he'd returned to the IoM, presumably with his new partner).
Before I had the lead that he was from the IoM originally, (I now have a birth certificate from the IoM for his birth on the 27 April 1899) I had searched through the Indexes of Births for the UK, and found nothing, so I'd be very interested indeed to get further information from you about the William Alltree Lyall you've found born in Liverpool in 1900 - can you give some more detail about what you found please? Look forward to hearing from you.