Let's have a look.
If its normal supercruise was M 2.04 at 60,000'
And it had excess power available (it did-dry and in addition had reheat available)
And its aerodynamic capability was quite a bit higher than its normal service Mach number (it was – Concorde 101 flew at M 2.23 or 1,450mph)
One could easily posit a test scenario in which a gentle climb was initiated just by selecting an altitude on the autopilot.
The autothrottles would do the rest in terms of power. Climbing at a steady Mach 2.0, Concorde could likely gain a significant amount of altitude. Perhaps 5-10,000' very quickly.
But before we go any further, we need to clear up some definitions to address some of the other comments.
The term 'ceiling' or ‘service ceiling’ in a normal aircraft refers to:
The density altitude at which flying in a clean configuration, at the best rate of climb airspeed for that altitude and with all engines operating and producing maximum continuous power, will produce a given rate of climb. A typical value might be 100 ft/min (0.51 m/s) climb, or on the order of 500 ft/min (2.5 m/s) climb for jet aircraft.
Normal airliners don't operate at their service ceiling, as this requires max continuous power and that is just a no-no for fuel burn. Also, you can get very dead, as the Pinnacle Airways CRJ-200 crew did by playing at the aircraft's service ceiling. You are at the edge of the aircraft's aerodynamic performance.
The service ceiling of 60,000’ commonly listed for Concorde, does not meet the normal definition as 60,000’ was its normal planned cruise altitude.
It is more than likely that its actual service ceiling 'hard limit' was quite a bit higher than 60,000' as Concorde cruised at 60,000' every day for 27 years.
I know, I flew on her in 1999 JFK-LHR.
The 60,000' certification service limit was (I believe) derived from a window loss scenario. If one window was lost the ECS could maintain cabin pressure until the aircraft descended to 10,000'.
Remember at 60,000' the TUC is a couple of seconds, and you require positive pressure breathing.
As to speed, the service speed limit for Concorde was a heat limited-127C. On my flight we reached 1,350mph and the nose probe read 126C (Yes, I spent about half an hour in the flight deck talking shop with the pilots and the very busy Flight Engineer) As mentioned, in testing Concorde ran out past M 2.23 but for service life reasons, its service limit was 2.04.
With the available excess power, a climb could be made while maintaining M 2.0-2.04. No sudden pull up required. At that speed, your climb rate at even a shallow angle is going to be impressive.