The motors look incompatible
A record player motor often serves double duty as a transformer, since it already has a bobbin and a steel laminated core. This transformer functionality is matched to the record player's amplifier. The Garrard one seems to have a tapping for valve heaters, which will put out about 90V AC between itself and one of the other 2 wires (no electrical separation, due to the connection in common) whereas the BSR one seems to have a separate winding for a transistor amplifier. It probably will be putting out about 24V AC on the white wires, when fed with 240V on the brown and blue wires.
Work-arounds would be either (a) use a resistor or capacitor voltage dropper, or (b) replace the original valve amplifier with a new IC amplifier, reusing the existing controls and speaker and leaving the original parts in situ for possible restoration in future. It might actually not be as crazy as it sounds, especially in the absence of a suitable high-output crystal cartridge. If you end up having to fit one of the "Chinesium" red stylus cartridges, which have a lower output voltage than the old crystal cartridges, then a simple UL84 won't have enough gain to give you a decent listenable volume.
Still, we can cross that bridge when we come to it. I think for now, we really need to see where the other ends of those motor wires end up.