Only just seen this....
So much "wrong" information on here
Actually this is a common issue on Weber Ecus from the cosworth era.
What actually is happening is the ecu is powering up and the hardware by
default, powers the fuel pump (and other items) momentarially.
This is then stopped as the processor power up circuit takes charge as
the first job in the software is to reset everything to a known state. (OFF)
I.E. The hardware design isnt that good in this respect.
Also, there is a facility for the software to run the pump on start up but
on nearly all chips, this facility is set to near zero values making it obsolete.
So to sumarize, it isnt a problem you have got its pretty random and normal
and will vary from car to car.
As for the car starting quicker, that will be true as the fuel pressure will subside
when the pump hasnt run for a while so takes time to build up.
Also it depends where the engine stops in its cycle of 720 degrees (2 crank rotations).
It could take up to 2 revs before the ecu even thinks about firing an injector !!!
The best way to start a weber cossie is to turn it on and start it immediately until it
fires up without any throttle.
(Assuming all is working correctly - especially the ISCV)
Hope that explains things well enough