Jack, I suspect there will be no centralised list of LS&GCs awarded to a particular Regiment. The medal (in its different forms) is awarded as and when an individual qualifies for one and so it something the Regimental Manning and Record Offices routinely monitored (today this is done by the Army Personnel Centre (APC) in Glasgow). Once a man qualified for a medal his details were passed to the Army Medal Office who arranged for the inscribing and dispatch of the medal and ribbon to the man's unit. The medal is usually awarded to the man (and now women) on a formal occasion such as a parade, often when a senior officer or dignitary is visiting the unit.
So a record of the award would have been held in two places: on the soldier's record card and at the Army Medal Office (as was).
LS&GC awards are not routinely notified in the London Gazette.
Since the Wiltshire Regiment hasn't existed (in that name) since 1959, I doubt if making a Freedom of Information request to the APC is going to be fruitful. You may have better luck FoIing the
Defence Medal Office.
I assume you are only interested in regular soldiers with the Wiltshire Regiment. The TA etc had different awards for long service and efficiency. Also you may need to differentiate between the LS&GC awarded prior to September 1930 when it was known as the
Long Service and Good Conduct Medal (sometimes with the word Silver at the beginning) and after 1930 when it was known as the
Medal for Long Service and Good Conduct (Military). The abbreviation LS&GC applies to both.