http://www.legionscotland.org.uk/u/f...rilMay2014.pdf
This may be of interest. The article attributes the Scotish cadets badge to those Schools who had a Cadet corps but which were not part of a JTC (formerly OTC and had their own bespoke badge design). However there were at least 4 School OTCs wearing the Regulars' badge for the Royal Scots, HLI and Cameronians in 1937.
It is interesting to note that none of the School Cadet Corps had Scottish Infantry regt affiliations and did not wear the Infantry badges of the day: hence the need for a Scottish Cadets badge.
It does not appear to have any London Scottish connection at all.
The text:
"As a school-based organisation, the Scottish Cadets existed from about 1935 until it changed to the Army Cadet Force during the War. The Scottish Cadets were attached to a number of schools, but not to public schools which ran the Junior Training Corps (JTC). There were also no regimental attachments in the Scottish Cadets at this time, nor did it have any direct contact with other Scottish Cadet units. The uniform was similar to Scottish Highland Regiments but with its own silver 'Scottish Cadet' cap badge. It included a jacket with brass buttons, all to be polished, a Black Watch kilt, a long horsehair sporran, tartan hose