The official map is plain horrible to use, and although they do have the exact route for all days except the last two (which are yet to be revealed), there's no way to get a proper list of streets or locations, or export the route to, you know, actual map/navigation application.
As to why I need this -- I challenged a buddy of mine, a London-ese, to snap pictures of as many torch relay runners as possible, on the condition that I provide him with an accurate street-level route plan. On the line are a number of beers, one for each photo, bragging rights, and of course, a handy conversation starter ("Did I tell you about that one time I followed the torch relay for a bet?"). So give a brother a hand!
I did some searching, and what I found are this this unofficial OpenStreetMap overlay and a nicer-looking interactive map found in The Guardian. I'd be happy with a detailed map of London only (say zones 1-6).
To preemptively address close-as-duplicate votes, there's this related question, and while somewhat helpful, it is not what I'm looking for.