All the examples of the real Home Forces wartime sign that I have recorded are of the small size you show in your first post. Also the red and blue are darker and the surrounding circles are close together and don't have the wider gap shown in the UKLF version. The attached image shows the two together.
I would suggest that 37 Signal Regiment's use of the UKLF version implied that at some point it was a unit commanded by GHQ UKLF. The Army Dress Committee was very strict about not allowing formation signs to be used as regimental dress distinctions.
The attached picture shows the sign worn by an unidentified Signals unit.
Hope that helps,
Jon
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