The phrase was chosen by Rudyard Kipling (whose son, John, fell at Loos in 1915).
Kipling was greatly affected by the loss of his son (having pulled strings to get him into the Irish Guards after he was rejected several times due to poor eyesight) and worked as the literary adviser to the War Graves Commission dealing with the setup of the numerous cemeteries in France and the inscriptions needed.
As such, he chose the words that would be carved on each gravestone with "Their Name Liveth For Evermore" (from Ecclesiasticus 44:1-15) if the soldier was identified, 'Known unto God' if he was not (his son's body was never identified). He was also asked to provide the words for the Cenotaph in London in 1921 and chose "The Glorious Dead". The cenotaph was ultimately used as a model across the Commonwealth alongside Kipling's phrase.

EDIT: There are also some interesting aspects to the war memorials in that the debates in parliament about the War Graves Commission and how it should operate and set up cemeteries was robust. There was a good deal of feeling regarding the uniformity, the lack of repatriation and the mixing of ranks. This 1920 debate in Parliament is most interesting reading.

Sources: Imperial War Graves Commission
"The Graves of the Fallen" Rudyard Kipling Imperial War Graves Commission, Annual Reports

Edit: added update link to sources and additional note on parliament debate.