The change over to Mk 7 cartridges was about 1910, and the Martinis soldiered on in the colonies for many years later. The change over was mainly to a spitzer and lighter bullet at a higher velocity commensurate with chamber pressures. And the Mk 6 cartridges disappeared. If the Mk 6 disappeared why would the Martinis remained in service?

The 577/450 was a black powder relatively low pressure cartridge.

I would seriously doubt if the Brits would convert the Martinis to 303 without checking chamber pressures, and use said converted Martinis with Mk 7 ammunition if it wasn't safe. They would also have ensured that all knew if the Mk 7 wasn't to be used in converted Martinis.

However, check your Martini to ensure it is indeed safe to shoot.