The table below show the badges listed in the 1937 Coronation Review of the Fleet at Spithead by HM King George VI on the 20th May 1937, but are known to have been in use earlier in the 1930's.
Physical Trainers
Alma Mater - HMS Temeraire, Pitt Street, Portsmouth.
The 1953 Review programme showed the Physical Trainer badge like this.
The badges below belong to the Gunnery Branch, the rules below are relevant to the Physical Trainer branch. By substituting the Gunnery badge with the Physical Trainer badge, you can see what the badges looked like.
Addition of Crowns and Stars to Branch Badges
In the 1977 Review programme, again, just this one badge was shown.