Council tax rates.

I should add the house prices were as at mid 1970’s valuations.

We live in a band C terraced house in London and pay £1,564 a year.

It doesn't matter when the houses were valued, it's just to place a house in a band. The values change constantly but usually proportionately over a district. If your house was valued at twice that of one elsewhere in your district many years ago the chances are the other house and yours are still at 2:1 ratio unless you or they have extended.
Under the old rates system your house had a rateable value which served the same purpose, but it could have been only £100 but the other house in our discussion would have been £50. The way the rate was arrived at was to multiply your rateable value by a pence in the pound constant. It was usually as much as £20 in the £ making yours £2,000 p.a. for example.

I live in a band C semi in a top ten deprived area and pay £2300.

I moved out of London to a market town in Bedfordshire and pay quite a lot more than I was in London it is a bigger house but even so I'm in the country now I dont understand it.

London rates are very low in relation to value residents say it’s to make up for high cost of living
Others say GLC is awash with money from congestion charges

This explains why council tax differs so much in different London boroughs.

London Council Tax Rates 2025/26 — All 33 Boroughs | Council Tax Checker share.google/FtVDNd0NwzAOqyVNY

My band D is £600 more than Richmond and they have far more wealth than us it’s a rip off

We pay £3376 for a Band E. I'm not convinced that COL has anything to do with it - I thought that council areas with a lot of people not contributing (eg students and under 18s), those in need of things like free social care, or people living alone and getting a discount had to charge more to cover it, with adult social care taking a huge amount.

Maybe the new care system will change that when it comes in?