grosso
The ecu only starts the fuel pump when you are cranking the motor your fuel pressure in your fuel rail is low as it is leaking back through the return line. So the fuel rail is not up to the required pressure. so replace fuel regulator as this is the return point for your fuel to your tank. That why it start the second the as there is the right PSI in your fuel rail. For example after market ECU will prime your fuel pump 3 -5 seconds be before cranking so your fuel pressure is up to the required PSI. If do not want to replace fuel regulator, then add a prime button to FC output of the ECU, press for 3-5 seconds or you can change it by connecting an out put that is triggered only when start key in on at where ignition lights come on. This will have fuel pump running all the time when ignition lights come on.

Thanks for the insight, 92 Hard Top. I will do some research over the weekend and decide how to proceed.

I actually followed this procedure last week. Page EG-217 of the workshop manual. Jumper +B and FP, turn ignition switch on. I could hear the fuel returning/recirculating, felt the fuel line above the fuel filter. Removed the jumper and started the car and it stumbled. I will do the procedure again tomorrow morning to confirm the result.

Falcon

I doubt it's a lack of fuel pressure as my other car which only switches the pump on at cranking puts up about 45PSI. in about a quarter of a second and fires up immediately to run at about 1500 RPM cold.
 

I have a fuel pressure gauge mounted on the fuel filter outlet, and have observed that my '90 model (gen 2 NA motor) seems to work the same.
Ignition on - nothing.  Turn the key and 40+ psi is there instantly, but AFAIK my car has never stumbled on the first cold start. 
 

Reddtarga

Falcon

I doubt it's a lack of fuel pressure as my other car which only switches the pump on at cranking puts up about 45PSI. in about a quarter of a second and fires up immediately to run at about 1500 RPM cold.
 

I have a fuel pressure gauge mounted on the fuel filter outlet, and have observed that my '90 model (gen 2 NA motor) seems to work the same.
Ignition on - nothing.  Turn the key and 40+ psi is there instantly, but AFAIK my car has never stumbled on the first cold start. 
 

It is not a spark issue, base on what I read here or air flow issue unless there is a faulty MAF which does make sense if it starts the 2nd time, Engine need air, spark and fuel for it to start. The ECU control cold start fuel mixture for starting up which is called open loop. It's not the Oyx sensor, that has to heat up first and this happen only when the motor is running. So it has to be a fuel issue. Quarter of a second fuel pump prime when cranking would be fine if there is no PSI lost in the fuel rail. remember the fuel regulators in these car are 20 to 25 years old. The cold start injector only give fuel in a pre crank starting then it straight to the main injectors. For example I'm installing a links I will not use the cold start injector as suggested by Links tuner. I will prim the fuel pump first so the fuel rail is at 45psi. Then the injector duty cycle pulses it with the first crank enrichment fuel, then post start enrichment and onto warm up enrichment. This only works correctly is fuel pressure is right, other wise your ECU fuel table will be lean. This will cause a first startup to stumble. If it is a gen 3 could be a faulty Map sensor which give a difference table reading for the ECU to get it info from.

Thanks everyone for the input so far. Still looking into the fuel reg, looking into replacing the vacuum hoses, also going to find a vacuum sensor I can borrow for testing. Want to eliminate a few more things before I start looking at the fuel pump.
 
The annoying thing about this issue is the car started OKish this morning after sitting for almost 18hrs. I started it went to 500rpm hesitated for a split second and went up to 1500rpm.
 
 

I had a fuel pump fail in a Commodore and it failed slowly and mainly didn't work well on cold mornings. It failed completely eventually and the intermittent start issues disappeared when the replacement pump was fitted.

Falcon
I would take a lot of convincing to it being a fuel pump. I fitted a new fuel pump (genuine from Toyota ).to my
1998 Bathurst a very few thousand km. ago. It still stumbles on rare occasions at cold start but has perfectly good
fuel pressure.

What prompted you to replace your fuel pump?