Angela - what else do you know about this chap?  And where did you get his attestation papers?  
There is something a bit odd about it..as he seems to have enlisted twice. Once aged 17 (born 1873) in October 1890 - and then again aged 20 (born 1870) in December 1890. 
Slightly different occupations and place of residence - but he is given the same army service number so is presumably the same boy.  (see below)
Did you know this - and have you taken into account the possible earlier (presumably false) birth when searching for his death?
Do you know what made the family think he had deserted? 
Also - I agree with candleflame - how do you know about his wife in 1905 being a widow?
John Connell O'BRIEN 
Born 1873    Islington, London 
Age at attestation: 17 years 10 months
Attestation date: 20 October 1890
Attestation corps:3RD BATTALION EAST KENT REGIMENT   
Number: 3548  
Address 40 Plumstead Road, Woolwich
Occupation Painter and Paper Hanger
(Employer/Master .. Mr J Bush)
John O' Brien  
Born 1870   St Albans, Deptford, Kent
Age at attestation: 20 years 6 months
Attestation date: 9 December 1890
Attestation corps: 3RD BATTALION EAST KENT REGIMENT   
Number:3584
Address 69 Hail Street Deptford
Occupation: Plasterer
(Employer/Master.. Mr J Emmett)
Milly