What would have happened when the Concorde's low-pressure emergency system brought it lower mid-flight?
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%.