its not likely to be the fuel pump but an easy test is to connect a piece of garden hose onto the test nipple on the fuel rail and put into a 2lt coke bottle or similar.
Replace the DME relay with a 3x way bypass (see Clarkes Garage Workshop manual)
With the ignition
off connect terminal 30 of the DME relay socket (always live 12v+) to terminal 87b which feeds the fuel pump. Let the pump flow for 15 seconds then pull the bypass out.
NOTE ( connect terminals 87 & 87b first to avoid potential shorts or other excitment)
Measure the fuel in a measuring jug (initiative required ie Wife's fluid measure jug) multiply by 4x and you'll have the pump's flow rate for 1x minute . From memory its likely to be about 2lts + but you'll need to check( edit 1350mls thanks JDS )
Next you should measure the fuel pressure which will eliminate the FPR (fuel pressure relief valve) Probably about 3 Bar but check .
Check to see how much pressure drops over next 30 minutes (should be about 0.5 Bar or less) this will eliminate the FPR and non-return valve in the pump both of which maintain fuel pressure.
From the symptoms you describe it sounds like it is temperature related so i would check the engine temperature sensor which controls the ECU also check to see that the fuel tank breather is not blocked and building up a vacumn. (test with the fuel cap off next time )
I'd locate the special ground points for the ECU and DME relay and the Fuel pump and clean those. Good earths are always a good place to start on 20+year old cars

The crank sensor is again unlikely to be at fault. Its only a magnet with a coil of wire wrapped round it. Easy to check the coil's resistance with a multimeter (refer to Clarkes garage guide) its more likely to be the the sensor's electrical connectors corroded or oxidised contacts or brittle wiring due to heat .If your tach needle jumps a little when cranking the chances are its working ok.
The ECU gets its power from the DME relay terminal 30 when ignition switched on. If the ECU doesn't see about 225rpm when cranking it won't energise the DME relay's second coil to supply 12v+ to the fuel pump. It also won't provide a pulsed earth to your ignition system.
Remember the pump only runs for about 1/2 second when you switch ignition on. Just long enough to pressurize the fuel delivery system. Its not until the ECU sees the correct data input from the AFM and Engine temp sensor and 225rpm from the crank sensor etc that it will actually provide earths for your fuel pump and ignition system so you get a spark and fuel.