I have recently taken an interest into the first World War I and its aftermath, and this one question I have yet to find a definitive answer for. I am aware of the various psychological problems and other health problems that soldiers returning home faced, but I have yet to find anything about suicide. This question is open to really any country that participated, but I am mostly interested in Britain.
I can answer in regards to Australian First World War veterans but but I would imagine that due to the widespread psychological issues experienced by soldiers from all countries involved on the Western Front, similar tragedies would have played out in Great Britain, France, Canada etc.
Upon returning from the war, many Australian servicemen attempted to return to their pre-war lives but the psychological impact of the war led to strained, often abusive marriages and family lives, rampant alcoholism and substance abuse. Many men often found it difficult to work, their PTSD made it difficult to foster social relationships and professional relationships and this in turn had a marked affect on their self image. Both pre and post war Australia were very masculine societies and being unable to work left many men feeling as though they could normalise up to theoreticians men in their families and communities. This led to more alcoholism and this in turn often led to financial difficulties. The culmination of all these factors inevitably led to depression and societal dysfunction compounded by the already often crushing effects of their war experiences. Unfortunately, as is still the case with current PTSD sufferers, suicide was often the last step for these men.
Pompey Elliot, one of the better known of Australia's First World War soldiers, was one such man who took his life as a result of his war experiences. Elliot began his service as a Lieutenant Colonel, commanding the 7th Battalion, and served at Gallipoli where he was wounded in the foot. He went on to serve on the Western Front, promoted to Brigadier General and he was present at the disastrous Australian attack at Fromelles where he was seen weeping in the Australian trenches after his men had gone over the top. Elliot served throughout the war on the Western Front and returned to Australia a highly decorated officer and was promoted to Major General in 1927. Post war, Elliot was diagnosed with "a nervous disorder" and ambiguous term for what we now recognise as PTSD. In 1931, Elliot attempted suicide by sticking his head in a gas oven and was admitted to a private hospital as a result. Elliot then committed suicide while an inpatient.
I'm out at the moment but I'll post my sources when I get home.
Sources: Homefront Hostilities: The First World War and Domestic Violence by Elizabeth Nelson
The Great War by Les Carlyon
Pompey Elliott by Ross McMullin