@emmathedilemma
I've always understood it to be to cover the higher costs of living but then other cities have high costs of living (admittedly probably not as high as London) and we don't get an uplift.
That's always been my understanding, too, and I agree with you about other areas being as expensive as some parts of London.
My house in Sussex is now worth 25% more than the one I sold 30 years ago in Croydon, but back then they were worth the same amount. And my council tax is £300 a year more.
I get where your employer is coming from though, OP. If you wfh, you could move to a much cheaper area and be quids in. I think you need to decide what's more important to you, your health or the money. And look at economies you could make if you really want to to continue to wfh.
Also bear in mind that losing £1kpa gross only results in a net pay reduction of £680 (less when you factor in pension contributions or are a higher rate taxpayer), so the financial hit may be less than you think.
Might your employer be prepared to offset the saving from you wfh a little by paying an allowance towards the extra costs in energy, water etc? I used to get £25 a month plus internet and line rental paid when I worked from home in a previous job. You might be able to recoup a bit that way.