I can add little to Bellerophon's post, except to expand on the fact that it's necessary to consider that the Vmo on Concorde did a little more than on a conventional aircraft, as evidenced by CliveL's earlier post.
You will also note that it starts at about 300kts before it increases to 380-400 (depending on current mass). I don't know this for sure, but it was explained that this was partly or wholly a 'bird speed', as some other aircraft have. That may be fiction.
It certainly didn't function as such at Nairobi, where one could happily encounter the sizeable local avian fauna at the 'full' 400kts owing to the elevation.
As a pilot this didn't consume much brainpower- you just flew exactly on the 'barber's pole' as much as you could while fitting in with ATC requirements.