There could be three sources for this sorts of photograph.
The first is a photographer paid for by the War Office who then distributed to copies to the relevant local newspapers for publication, in which case the photograph would have been subject to Crown copyright. It is possible, although I think it's a very slim chance, that the original may be archived somewhere in the National Archives.
Added. It is also possible that his Regiment arranged for a photographer to take the pictures, for publicity purposes. If that was the case, the Regimental Museum might be the best place to look for any surviving copies.
The second possibility is that the local newspaper sent their own photographer round when the soldier was home on leave, possibly to attend an award ceremony in his regiment's home town. In the latter case the copyright would belong to the newspaper and may have survived in their archive. Even though the newspaper and its publisher will probably be long gone now, it is highly likely that their archives have survived. These things were bought up along with the newspaper titles, by other publishers, and you can often trace their lineage forward to companies like
Archant or
National World plc today. They probably won't have maintained the physical archive of photographs from a century ago, and will have transferred them to a picture library such as
Gale or the
Mary Evans Picture Library where they continue to be monetised. The local county archives is also a possible place to search, but given that newspaper archives had real economic value they tend not to be just given away to a public archive.
And the third possibility is that it is a photograph which the family supplied when asked by the newspaper. However, looking at the newspaper photo of your grandfather, I don't think that was the source of his photograph. The style looks editorial rather than social.
