[SteveSarre]

Hi BlueStag

As Baffler says, you shouldn't just assume that it is the pump.

If the pump buzzes for a few seconds when you turn the ignition on (you may have to get someone to listen under the boot), then the pump and relay may be fine. It turns itself off when the necessary pressure is reached.

If the car runs fine when it has started, that would also tend to indicate that the pump can keep up with demand, (and that the relay is fine).

If the immobiliser system is not satisfied that a correct key is in the ignition, the pump will run etc as normal, but the engine will be shut off after just a second or two. ie the engine will fire up as normal but then quickly shut off.
The chip in the key is detected by an antenna ring (which sits around where the key goes into the steering lock). If you have a second key you should try that to see if that helps, but the antenna ring is more likely.

I think a code reader would tell a mechanic what is going on.

HTH
Steve