I guess understanding the changes first entails using the correct names for weapons. Canada did not use the Sterling, we used the C1 SMG which was a Canadian Arsenals Limited manufactured 9mm SMG with Canadian features not found on British manufactured weapons. Calling a Canadian C1 SMG is like calling a Land Rover a Jeep. The 9mm Mk II Sten was replaced by the C1 SMG around 1960.
Rough dates for the change from .303 Bren to M1919A4 .30 cal - is early to mid-1960s as at this time the 7.62mm FNC2 was also replacing the Bren. The copy I have of C-30-541-000 CJ-001 Car, Armoured, Ferret, Mk I Scout Car, Command Car, Missile Car .30 Browning Machine Gun Kit Installation Instruction is Issue 1 from Oct 1965 so I guess that would be a good indicator of when the official change came in for the Ferret although I have photographs of M1919 armed Ferrets that pre-date this instruction such as this 1964 Cyprus photograph.
As Rob said, the 7.62mm C1 GPMG began in the late 1960s with the C1A1 7.62mm GPMG coming into service about a decade later or just about the time the Ferret was going out of service.