There is one William Roe who served in WW1 whom I haven't been able to eliminate so far, so just to note him for now. I can't see a service or pension record for him, but there are three separate documents for him under the regimental number of 341980:-
(1) a medal card for William A Roe, showing him in the Labour Corps, enlisted 31 July 1916, discharged 14 September 1917, cause of discharge 'Wound', address = 56 Morse Street, Swindon.
(2) a pension ledger index card for William Roe who was discharged on 4 September 1917, in the Labour Corps, birth year indicated as 1899, record type = disability, and his residence place given as 'Post Office Salford Manchester'.
(3) a Silver War Badge record for William Roe, enlisted 31 July 1916, discharged 14 September 1917, discharge regiment = 5th Royal Irish Rifles, discharge unit = Labour Corps, cause of discharge 'Wound', age at discharge = 19, served overseas.
Edited to add: There was an article in a Bath newspaper of May 1920 that has a resonance in this case, under the title of TWO DESTITUTE EX-SOLDIERS, it is about the soldiers wrongly receiving 3s 6d from the Bath Branch of the Discharged Soldiers' Federation, for which the Bench took a lenient view, bound them over to be of good behaviour and arranged to send them from Bath back home to London. One of the soldiers was described as 'William Roe (21) ... a music hall artiste ... 13 Tressillian Road, Lewisham, S.E ...'.
It goes on to say 'Roe was discharged from the Army as unfit for further service as the result of a gunshot wound. His pension had been discontinued, but its renewal was under consideration. Apparently his Army record was good ...'.
Edited again to add: Just found another two pension ledger index cards for William Roe, definitely the same person as mentioned at (1) - (3) above (regimental no. 341980... and other details match), one card shows residence as Bristol, the other shows residence as Dublin.