In addition to the example on this forum, I have seen two of these offered for sale in auction in the UK and both have been described by the vendor/auctioneer, as 'billiard scoreboards' although that doesn't in itself make the description correct.
I know that Burroughes & Watts first introduced an electronic scoreboard in 1875 which worked by buttons set into the cushion rail of the table, connected to a battery stored under the table, and wired directly to the scoreboard on the wall. This scoreboard seems to have the same connectors to allow this arrangement.
The only issue I have is the scale, as games of English billiards from this date were normally 100-up, and the Burroughes & Watts dials used this scale, albeit that the dials were significantly bigger. It could be that 50-up was selected to make the score more easily seen with the smaller dial used by Shimmin.
It is not clear exactly how the gearing works for the smaller dials, but multiples of 50 would not work for billiards, which is an issue that works against my assumption. I have also been unable to find this item being advertised for sale in the local area or elsewhere, although if he was directly supplying local billiard rooms, I would not except them to have been advertised for wider retail.
I'm still looking for conclusive evidence on this one.