As I see it, you can, with the Kindle, choose between paying, after free introductory periods, £9.99 a month via the Amazon.co.uk web site or £2 per week for internet access. With internet access, you would want to set up a scheduled download, using your Times logon and password, with free software available at
http://calibre-ebook.com/.
Most people would go with the Amazon version as it likely is more complete and doesn't cost much more. However, if you have the 3G Kindle, and it works as in the states, the £2 option gets you most of the paper automatically every morning via Wifi in your home, and you then could check TheTimes.co.uk during the day via the browser, or could get repeated Calibre downloads if you have WiFi access at other times.
If you are independently wealthy, you could always subscribe to both