Why no flaps?
Flaps change the pitching moment of a wing. After all, they add lift over the full chord, so the sum of the increased lift attacks at about mid-chord, which is a quarter chord aft of the regular lift. If there is no separate tail surface to compensate for the pitching moment caused by that extra lift, the aircraft will quickly pitch nose-down and crash.
Next, flaps change camber and, therefore, are destabilizing the wing. Without a stabilizing tail, a cambered delta configuration will become unstable. The only camber which is helpful for delta wings is at the leading edge and must be compensated by a little trailing-edge up deflection of the control surfaces. Positive camber near the trailing edge is destabilizing and can only be tolerated on a flying wing with artificial stability augmentation.
Why no slats?
Slats are helpful for delaying flow separation to higher angles of attack and allow a wing to create more lift. To understand their effect, it is not enough to consider what they do to the flow around the wing, but also the effect of the wing on the slat needs to be understood. A slat is like a small wing flying just ahead, and therefore in the upwash, of the main wing. The wing will induce a very high lift on the slat and in turn see its suction peak around the leading edge greatly reduced, which helps in keeping the flow downstream attached. The comparison plot below should illustrate this effect nicely:
Figure 36 from A. M. O. Smith's paper "High Lift Aerodynamics"
But a delta wing at high angle of attack does not have attached flow on its upper side. It makes use of flow separation at the leading edge which creates a powerful vortex over the upper wing. This is called vortex lift. So for take-off and landing, deploying leading edge devices would help Concorde only a little - they are most helpful in the region just before vortex lift kicks in. This would be for subsonic cruise, for which the original Concorde wing was completely unsuitable. Adding camber at the leading edge would had increased subsonic L/D a lot, so the subsonic cruise segments (like all flight above land) and flight in holding patterns would had been much more efficient. That was not considered initially, and the lower complexity of an un-slatted wing was preferred.
With Concorde B, a span increase and the addition of variable leading edge droop (no slats!) were proposed. The picture below is taken from the original web site which has been the source for the site you linked to in your question.
Concorde B aerodynamic improvements (picture source)
If you now look how the L/D would had been improved by this, it becomes obvious where the leading edge devices helped most. The L/D improvement at take-off and landing, by the way, can be mostly attributed to the span increase.
Concorde A and B aerodynamic efficiency comparison (picture source)
This information is missing from the site you linked to, but is essential for understanding why Concorde B had variable droop added: It should help to extend range beyond the Paris-New York connection and allow more efficient subsonic flight (especially the hold at 250 kts).