Hi Jim,
A little it more information on the level at Garrigill.
According to the large Greenwich Hospital estate, plan, part of which I have a copy of, my plan, 52c, the level is described as a New Level, and presumably driven on Thorter Gyll Syke Cross Vein. This vein is shown running along that burn.
Unfortunately the map has no date on it!
I have also checked my plan 154, which is acopy of KCD ‘s wartime mapping on the area Garrigill-Nenthead, it just shows a cross cut, not unfortunately named.