I can’t speak with any authority on Concorde specific requirements, but it seems that changing the nose droop angle would be a fairly major configuration change. Below roughly 500’ AGL aircraft are generally considered to be either in the take-off or landing phase of flight, and manufacturers and airline operators prohibit configuration changes at low altitude.
Take-off phase – Take-off power and best climb speed are used to get safely away from the ground. With the exception of retracting the landing gear, nothing else but basic pitch control is touched. (roll control and rudder as needed to remain wings level, but no turns initiated) Above some predetermined acceleration height, (may vary by airline SOP, but 500 AGL is a good approximation) the aircraft is gently nosed over to accelerate to flap retraction speed. Around that time when the flaps are up the power may also be reduced somewhat to climb power for engine life and/or noise abatement procedures, and the aircraft may turn on course as it transitions from takeoff to climb phase.
Landing Phase – Below 1000’ AGL the aircraft should be ready for landing as the crew prepares to transition from approach to landing phase. For compliance with Stabilized Approach criteria, this requires that the aircraft be configured for landing, (gear, flaps, etc.) with the “etc.” in this case presumably covering the nose droop. Below 500’ especially the crew needs to be focused on instrument approach minimums and transitioning their scan to the runway environment for landing. I don’t know what sort of aerodynamic changes the nose causes, but it just doesn't make sense from a best practice standpoint to make any sort of change down low on a fast moving jet that is probably 20 seconds or less from touching down.