But it ain't that simple. It turns out that the word cookie has in fact come into British English, and even means something like what we Americans understand. But it hasn't replaced biscuit; instead, the two terms in BrEnglish now have a subtly different definition.
Lynne Murphy at separated by a common language explains:
In AmE, cookie refers to what BrE speakers would refer to as biscuits, but also to a range of baked goods that were not typically available in Britain until recently--what we can call an 'American-style cookie'--that is, one that is soft and (arguably) best eaten hot. Since in the UK these are almost always bought (at places like Ben's Cookies or Millie's Cookies), rather than home-baked, they also tend to be of a certain (largish) size. In BrE, biscuit retains its old meaning and applies to things like shortbread, rich tea biscuits, custard creams and other brittle things that can be dunked into one's tea, but cookie denotes only the bigger, softer American import. (In fact, twice this year I heard Englishpeople in shops debating the definition of cookie, and had noted this for further discussion on the blog...and here it is. For previous discussion of this and other baked good terminology, click here.)
The semantic range covered by a word is rarely fixed for all time, and the language adapts to address new circumstances, to accommodate new terms, and so on. In HistLing101, for example, they'll tell you that the English word deer, which refers to a particular type of ruminant, is etymologically related to the German word Tier, which simply means "animal." As the West Germanic languages split apart, what was once the same word claimed different semantic territory in the descendant languages. In the case of biscuit/cookie, in British English, a new thing was introduced that came with an existing name. One way to accommodate this would have been to also refer to American-style cookies as biscuits. But clearly British speakers have felt that it's useful to distinguish this new type of baked good from another type, enough so that British English now has two words for these different things. In the process, the definition of biscuit in BrEng has become a tiny bit narrower, inasmuch as it now does not cover all the territory that is covered by what Americans call a cookie. In those lists of American vs British terms, the line for cookie=biscuit should now at the least have an asterisk and a note that makes this point. (Not that any such note will likely be added.)
American speakers continue to use cookie to cover the whole range of baked goods here, so they don't yet feel that the difference requires a split in terminology. They sure as heck would get confused if BrEng speaker started talking about these subtle distinctions between biscuits and cookies.
I will note for the record that I'm aware that within the world of retailing (and baking), there are all sorts of different names for styles of cookies, even in (especially in?) AmEng -- ya got yer snickerdoodles and fig newtons and vanilla wafers and ginger snaps and animal crackers, all of which refer to specific recipes or styles of cookies. I think, however, that Americans would still consider all of these to be cookies. Yes?
 I worked in the UK a while back, and ran afoul of such a difference on my first day (!). My employer issued a card that could be used to pay for fuel for the company car. I was told to go to someone and get such a card; when I did, I asked for a "gas card." I got puzzled looks until one of us (probably not me) sorted out that what I wanted was a petrol card.