Hi
Welcome to Roots chat, there are so many helpful folks in here I am sure someone will help. A cautionary word though, we are not allowed in this open forum to enter / search living people as the 100 yr data protection applies. However after you have posted a couple more messages you will be able to send personal messages in the event your search begins to include living people.
I can see your family in the 1911 census- have you seen that?
It is interesting and probably a coincidence that your Mums, Grandmother who was Margaret McClay was also from Scotland - Paisley, Renfrewshire and she was B circa 1835. She was the head of the family in Salford in 1911
In case you haven't seen the census it also shows William Taylor B circa 1876 Dumfries and his wife Margaret B 1881 Liverpool. They had 2 daughters with them Agnes B 1905 Salford and Mary B 1907 Salford. The family were living at 8G Industrial Dwellings, Greengate, Salford.
So that's a start. You may need to go to the Scottish Records Office and in there for a fee you will be able to search and see the actual birth certificates for both Margaret Snr and William Taylor.
Now to his supposed Military service, at the time war broke out in Aug 1914 William would have been aged c 38. He was most likely living in Salford and may even have joined up as an escape - many men did expecting the war to offer excitement and a chance of travel - far from their pre war hard and often dangerous work / life?
I will have a look at Military next and see if we can pin him down.
NDRFT