We do still say 'biscuit' in normal speech and our supermarkets, even the one owned by Walmart, still have biscuit aisles. However we are widely exposed to American culture and in the context of sports commentary "raiding the cookie jar" does work better than "raiding the biscuit tin": somehow "raiding the biscuit tin" just feels much more petty. This is perhaps because, although there is an overlap, an American style cookie tends to be bigger, softer and generally more indulgent than a British biscuit. In fact large, soft, special treat biscuits are sometimes sold as 'cookies' although they are usually so large we would probably not think of them as a 'biscuit' anyway.
Having said that Americanisms do appear in proper English, especially when promoted by American companies. Following the penetration of the British market by McDonalds and Starbucks most vendors of coffee, including iconic British brands, now offer 'regular' cups rather than 'standard' or 'medium'. That is unless they aspire to Italian sophistication and call them 'medio'.
One area where you will hear 'cookie', though, is computing. Small files downloaded to your computer by websites are always called 'cookies'. I can't imagine any British person wanting to change that. In fact most users probably don't even make the connection between leaving the file and presenting a child with a biscuit.