Hi Mark;
Reference: 1st, The Queen's Dragoon Guards - formed in 1959 by the merger of the 1st, King's Dragoon Guards and The Queen's Bays (2nd Dragoon Guards) - Recruiting Area: - Wales, Herefordshire and Shropshire. Home Headquarters is in Maindy Barracks, Cardiff, Wales.
Reference: Carabiniers - 3rd Carabiniers (Prince of Wale's Dragoon Guards) merged with the Royal Scots Greys (2nd Dragoons) in 1971 to form The Royal Scots Dragoon Guards (Carabininers and Greys) - Recruiting Area: - all of Scotland. Home Headquarters is in Edinburgh Castle, Edinburgh, Scotland.
Yes, I agree with Mark, there have never been any Cavalry Regiments with the word 'Welsh' (or 'Welch') in them, in the UK Forces.
Cheers