Interesting, Looby! Could be him! And I found this, based on that find:
on ancestry, British Army WWI pension records/ William Reid b. abt 1871 in Milson (mis-transcription for Neilston?), Renfrewshire/ married 22 June 1901 in Glasgow to Elizabeth Alexander/ document year 1914/ Regiment name: Princess Louise's (Argyll & Sutherland Highlanders)/ Army Reserve (special reservists) Attestation/ children: William, Kenneth, John, James & Elizabeth.
Is this the same William that says his parents were William Reid and Elizabeth McVey?
K.