To comply with regulations since 1955 at least, only lamps with a filament bulb could be without an earth, and since 1966 only pendents. As to safety with a class II likely safe enough, but any inspection is likely to list it as a fault.
I'd be more concerned with having the electrical installation inspected and tested rather than fitting different light fittings. Not sure I've ever seen cable with only red and white cores, except for speaker cables.
I have, it's a flexible, which suggests there must be some sort of joint above the ceiling, to the fixed wiring. There might be an earth in the fixed wiring - really it needs to be investigated.
(ix) Lighting fittings using filament lamps installed in a room having a non-conducting floor, mounted at such a height that they cannot readily be touched and are out of reach of earthed metal.
A circuit protective conductor shall be run to and terminated at each point in wiring and at each accessory except a lampholder having no exposed-conductive-parts and suspended from such a point.
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