I have two portable record players here that date from a similar period, one with a 2-ire and the other with a 3-wire cord (these are 200 to 250V units).
The ungrounded unit doesn't pose any shock risk, although by gently touching the metal record deck with the back of a finger it's just about possible to feel some minute capacitive leakage.
I've seen a couple of schematics for old U.S. equipment which actually tied the chassis to one side of the line via a 1 meg resistor. To the best of my knowledge, that was never done on British equipment, although some did couple with a low value capacitor, say 0.001 uF or less.
