Apparently the pair of engines on the left was burning. And that is why there was a long streak of fire behind the aircraft. The fuselage was not burning. It was the engines. I think that apparently the pilot on the flight data recorder and the cockpit voice recorder will almost certainly indicate that they struggled valiantly to get the aircraft under control to land somewhere and lost control at the last few second.
Rotation means lifting the nose off and flying.
The tower, if they indeed signaled the crew, it was at or about vr, v rotation, at which point the plane is committed to flight and would be unable to stop on the available runway. It would have to continue to fly, and that is apparently what happened. The Concorde, the cockpit, and its original design were analog. The Concorde is not an all-electric airplane. It has been modernized with many new avionics which are digital.