My understanding is that the Concorde was equipped with a system that automatically lowered the aircraft to a safe altitude if cabin pressure was lost. I know such emergency never occurred in its history but did the Concorde have the range to make it to the closest airport if say this happened in the middle of the Atlantic. With a range figure of approximately $6500$ km at Mach $2.0$ and $50000$+ feet cruise, it is not clear that this bird would make it to land in such an emergency given it was terribly inefficient at low speeds and altitudes. What happen if this emergency had ever happened?
I'm answering a point in a comment by the OP, rather than answering the question itself directly - but I do think it's totally relevant. Feel free to vote accordingly :)
This list provides a list of Concorde destinations. I am assuming that it is complete. I have not gone through every route but looking down the list suggests that GOOY-SBGL (Dakar to Rio) is the route with the longest ocean crossing. This list seems to confirm this.
Concorde did not fly the southerly Pacific routes, I guess because it would not be capable of diverting in exactly the situation you ask about.
Suitable diversions on that route would be FHAW (Wideawake on Ascension Island), SBNT (Natal), SBRF (Recife) and SBSV (Salvador).
Using this tool, we can see that GOOY-SBNT is 1870nm and it's all ocean between them.
Using this tool, we can calculate that the mid-point of GOOY-SBNT is at 10° 26′ 47″ N, 026° 29′ 44″ W. We can also calculate that the midpoint to FHAW is 732nm and the midpoint to SBNT is 587nm.
So a reasonable guess (without access to the original flight plans) is that the maximum distance from a diversion is to SBNT from the midpoint which is 587nm. Easily done, even at 10,000 feet.
Before the midpoint, I guess they would have returned to GOOY since it is closer than FHAW. Therefore, I also guess that FHAW would only have been considered if SBNT and GOOY were not available - most unlikely.
I have now found a longer one! KLAX (Los Angeles) to PHNL (Honolulu). This one's easy. The Great Circle mapper shows us it's 2220nm. There is nowhere to divert between those two so the new greatest distance is 1110nm.
Concordes' max range was about 3900nm so as long as it was about 60% efficient at 10,000 feet compared to cruise (1110 / ( 3900/2) plus a bit for go-around time), the diversion was no problem. Although inefficient at lower levels, I doubt it was less than 60%.
The pilots had a clear procedure to follow for an emergency descent, but this procedure was not automated by a machine. (It's actually quite complicated because Concorde had no speed brakes. So it requires negotiation with air traffic control for a clear descent path---not much point in doing an emergency descent if you crash into another aircraft on your way down).
Concorde was actually fitted with very small windows, so that a powerful air pump would be able to hold a pressure differential against the outflow in the event of one window loss.
Hear more from a Concorde pilot at: http://omegataupodcast.net/2015/02/166-flying-the-concorde/
There are several diversion airfields on the transatlantic route; KEF, YQX, and SJF are the most famous.