Alexander Grant didn't marry until 1918 in Scotland, just 7 months before he was killed.
Alexander Grant 30 Motor Car Driver (Corporal 3/5 Bedfords) of Crowborough Camp, Essex, son of Alexander Grant and Jessie nee Calder
married
Elizabeth Barbara Simpson 29 School Teacher of Logie Cottages, Forres, Elgin, dau. of Robert Urquhart Simpson, Postman (deceased) and Margaret Ann, nee Inch.
Married 29/1/1918 Forres, Elgin
So you need to be looking for an Elizabeth Barbara Grant (rather than Lizzie).
Can't see any trace of her in England records so would guess she returned to Scotland after Alexander's death.
Annette