Coveralls were De Rigeur on Excercise, & also round the workshops back in Barracks.
The job was dirty, crawling around & under vehciles would make a uniform grubby in no time!
BD's would have been worn during the fifties, yes.
When leaving the workshops & returning to the accomdation Block/ buildings. Coveralls would be removed & Normal Uniform worn.
Trade badge, are you sure it's on Green? Normally Khaki Material/felt.
Yes, worn a brassard like you had in the cadets.
BUT, Nver worn alone, Only with a Rank badge. L/Cpl, Cpl, S/Sgt. NOT with a Sergeants rank on it's own!
Staff Sergeant, Yes. But ONLY if an Artificer.