Could the Olympus 593 have been windmill-restarted during supersonic flight?
Yes they could be re-lit without cross-bleed (all engine flame-out above M1.2).

Regarding lack of APU, Concorde has emergency power generation based on hydraulic power (which will be operational above M1.1 due to windmilling and is used to power some or all the fuel pumps for the re-start). (If you're wondering about slower speeds, Concorde would use RAT.)

Regarding the spill-doors, they do not dump all the air out. The automatic operation is a function of inlet pressure (among other parameters). They will open to reduce pressure build-up (and may also open at low power setting above M1.3), but air will still enter the engine and exit the back end.

The all engine flame-out above M1.2 procedure (first 2 pages attached) is simple in nature and does not contain any notes on manual spill door operation. To summarize it:

Page 1:


Fuel transfer pumps will be configured for fwd transfer;
Hydraulic system will be configured for emergency power generation;
Avionics cooling will be configured (anticipating loss of air conditioning);
If no automatic re-light, then thrust levers to idle to permit manual re-light.


Page 2:


HP [fuel] valves (which feed the engines) to be selected off (so not to drown the engines);
Auto ignition off;
Select re-light mode;
Wait 30 seconds from turning off the HP valves (for fuel drainage);
Open HP valves and wait 20 seconds for a re-light.


And below is the optimum re-light envelope.





Source: Concorde FCOM