This one is being used as a cap badge on a Field Service Cap (the more normal configuration for the RA), but it is a standard collar badge mounted on a red cloth covered piece of tin for extra 'bling'. This is the type he has on his collar in the photo, as it's brass and shiny.
I posted a US-made Royal Artillery BD in a thread recently, but I took a quick snap of it to show you the collar. This one belonged to L/Bdr Fenton in the 1st Division in Italy (same theatre as Williams). It has definite signs that it had collar badges on too:
By the wear and general grubbiness of the jacket, I would suspect that if it was his 'best BD' it was probably later relegated to active use and the badges removed (by him). However, BDs with collar dogs is an odd practice anyway, that seems to be more common amongst Italian Campaign units than ones in other theatres.
Finally, I took some pix of some Artillery Field service caps for your general interest:
Top to bottom: 1. Royal Artillery ORs with brass collar badge. 2. Royal Artillery ORs with plastic collar badge 3. Royal Artillery Officers with chocolate bronze gun badge (this one actually worn by a Canadian RSM)
1. Royal Canadian Artillery ORs (cotton material) 2. Royal Canadian Artillery ORs (wool serge material)
And finally some nice dress caps. 1. Royal Artillery ORs Coloured FS Cap
1. Royal Canadian Artillery ORs Coloured FS Cap