Don't know what others think or whether I'm alone with these thoughts.
Hence the awaited details from the marriage certificate of your man being so important here.
No Annette, you're not alone. The William born in Chester-Le-Street may well be the correct person - particularly as there no birth registration for a William Cooper born Apr 1897 in Gateshead (per the details originally given) - but the OP does need to confirm via the marriage certificate that his father's details match up.
If he was born Apr 1898 and married in the Apr-Jun qtr of 1914 then he was only just 16 when he married. His enlistment in April 1915 states his age as 18 yrs & 4 months - it is only later that his DoB is recorded as 1 Apr 1898 'certificate produced'. Perhaps he lied about both his age and marital status in order to enlist? I see he was only 5ft 2in and under 9st. I know he wasn't alone in that, but it really hits home that he was little more than a child himself.
His record shows he went missing for 3 days in Jan 1918 and on 13 Feb 1918 was admitted to the 1/3 Northumbrian Field Ambulance as not yet diagnosed (mental), and then onto what looks like Casualty Clearing Station 62, also NYD M[ental]. Returned to duty under escort 21 Mar 1918 and then gunshot wound to leg (?) a little over a week later?
I'm no expert on reading these records so it's worth asking for help on the WW1 board, but as he seems to have gone awol a few times and then later walked out on his family, I wonder if he was actually suffering from what we would now call PTSD?