This link refers to SGN, who manage gas networks in Scotland and the south of England, but I'm sure that the same policy will apply elsewhere:
https://www.sgn.co.uk/Our-Services/SGN-Meter-Box-Repairs/
i.e. the meter box forms part of the property that it's affixed to, so it's up to the property owner to keep it in good repair. (While it's possible that a rental agreement might seek to place responsibility for such repairs upon a tenant, it's unlikely to do so and probably wouldn't stand up in a court of law anyway).
So it appears to be the landlord's responsibility.