Those banners (in most cases) are not you turning on or off cookies. The bbc sets cookies immediately (they do allow you to unset some here http://www.bbc.co.uk/privacy/cookies/managing/cookie-setting...), but you already have a session cookie that will last 4 years once the page is loaded.
Most sites will state they are assuming consent based on your continued usage of the site.
Most sites will state they are assuming consent based on your continued usage of the site.