A company can’t just ‘adjust’ an existing employee’s salary without consultation and agreement, that’s contractual variance.
True London weighting — where it is a specific, separate figure — is generally applied to jobs where people have never, generally, been able to work from home: teachers, police, firefighters, some civil servants. Mostly public sector.
For everyone else, it’s simply a higher salary that is offered or negotiated.
DH works in this field; the short answer is ‘it’s complicated’. While it would be shady — and potentially actionable — for a company to reduce someone’s salary based on their personal circumstances changing, some companies are looking to pay less for remote workers if someone on a higher salary leaves and needs to be replaced, for example.
But, to be honest, it’s mostly about the person and their perceived value to the organisation; if someone is seen to be worth X, and they ask for X and can back up their position, they get offered X.
Where they live has little to do with it, unless distance becomes an issue directly impacting performance and team coherence, which is an emerging issue.