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CYOA-
No, just cut off the EU plug and put a UK one on the cable - no soldering needed.
Could even get one of these if you want to make it easier
https://www.amazon.co.uk/LRMYS-Adaptor-European-Converter-Business/dp/B09PKS6VJD
spotter
It's in a plastic box, so unless there's a ground contact on the PCB (which there won't be if it was designed for a 2 prong plug), there's no need to connect an earth.
If it came with a UK plug it'd be one of those with a plastic earth prong.