The 31C was always known as the 31C. Your particular engine with prefix 051D was originally supplied to Excelsior from 1960-65 which was about Excelsiors last gasp. Lots of bikes and scooters fitted the 31C. Too many for me to sit down and list. The survival rate is pretty good as 148cc isn't a convenient size for competition. It doesn't suit the common capacity classes.
The standard Villiers 31C operators handbook and spare parts list is available from the usual sources. I use the VMCC Library myself. Look at their website,
www.vmcc.net and go through their library holdings.
In latter years Villiers made variants of their engines to suit various uses. Eg, fan cooling for scooter use and the Siba Dynastart for electric starting. The VMCC have all the info.
These will also tell you which carb and I'm sure one will appear on eBay once you know what you're after. Villiers stuff isn't rare. I would expect a piston will be available from the usual sources. Either Vintage Motorcycle Pistons in Shropshire or Red Lizard Classic Motorcycles.
Cheers,