Perhaps the British public realised that it couldn't be: On a recent BBC audience vote, the Underground map came second to Concorde (which could be a design icon of Britain, but probably not in the way that viewers intended). I even voted for the Underground map myself, but mainly as a cynical attempt to boost book sales.
If the Underground map is to be a design icon, the obvious question is, which version? The simplest answer is, whatever the current one is, but this (based on the 2001 revision at the time of writing) is held by many to be a really awful piece of design, and many of its features are objectively bad from a human factors point of view. If the current Underground map is an icon, it currently stands for the decline of British design standards!
Lets try a different track: Concorde is an obsolete aircraft, so perhaps it is OK to for an obsolete Underground map to be a design icon? The trouble is that unlike Routemaster Buses, traditional phone boxes, and Concorde, the Underground map has had a long history of major changes, with good designs and awful designs. The British general public were voting for something, but they did not know what it was (nothing new there). Overall, what they believed to be iconic seems to be a concept, that taking a complex rail network, and distorting it in an unspecified way according to some reasonably well-defined rules and principles, so that it is easier to understand, somehow stands for Britain. Not really what iconic means: the Mona Lisa is iconic, but Italian renaissance portrait painting is not, or at least it shouldn't be (try inserting your own favourite image as the icon, versus its genre, you get the general idea). So, can the London Underground Map be a design icon? Only if you change the meaning of the word.
Is any Underground map design particularly outstanding, and deserving the title of 'best of the bunch'? That's a really hard question, so I will pass on that one for now.