It still seems to me like it could be a fuel pump that is turned on late (such as a slow switching relay) or for some reason is slow to pressurize the fuel system. If the car has been recently driven (in the last hour or two), the fuel system still has pressure in it. Let it sit long enough to get cool, and the fuel system pressure drops as well.
A simple test without needing any equipment would be to:
1) run the car to warm it up (since you say it works when warm)
2) turn it off
3) pull the fuel pump fuse
4) try to start the car again (this will drop the fuel system pressure to zero)
5) put the fuse back
6) start the car normally (by normally, I mean turn the key from off to starter position immediately - don't let the ignition stay on for awhile before engaging the starter motor because that would give time for a slow relay or slow fuel pump to do its work, and you want to find if one of these is the cause).
Do this test a few times to make sure you have consistent results. If the car is slow to start after depressurizing the fuel system, it is a fuel pump or relay problem. If it starts right away, look for other causes.
You can also try the test with a cold engine (say, after sitting overnight) to gather more data.
By-the-way, we even have a fuel pressure delay related sympton on the Elise. If the Elise fuel system depressurized, and you turn on the ignition before pushing the button on the key fob to mobilize the system, the fuel pump does not turn on until you push the starter button. This causes a delay in starting. If you push the key fob button before turning on the ignition, then the fuel pump turns on when the ignition is turned on, which gives the fuel pump a few seconds to pressurize the system before the starter button is pushed.