Very interesting to know. Many readers may not be aware that council tax in the UK is quite regressive and tops out at ~£4-5K / year on properties valued higher than ~£1M. So you can own a £5M GBP house and still pay only £5K / year for an annual effective property tax rate of just 0.1%.
This is one of the reasons buying a luxury house in the UK is comparatively quite cheap in terms of total cost of ownership compared to many states in the US where you have to pay much higher property tax rates, insurance, and so on.
So even if the council tax is doubled on a second home, you still might be paying only 0.2%. Compare that to an average property tax rate of ~1.8% in NYC (before pied-a-terre).