Thanks for the information so far!
I did write to the DVLA, but they said they have no information on the registration number in the photo.
The "sliced" photo is as much of the motorcycle as was in the original (which was relatively short and wide). I decapitated grandad and took off the buildings and sky from the top - but that's all there was in the original photo. The second, rather scratchy photo has a more complete view of the motorcycle.
Looking at old photos on google, it seems to look something like the 1910s Norton Classic as at
http://www.vintagebike.co.uk/pictures/pre-1912-norton/ - judging by the springs and bends of the front forks and sloping front of the petrol tank. But the mudguards aren't the same (even between photos). But, then, a Triumph H (to my eyes) also has similarities.