Recently had the same problem with my Astra - exactly the symptoms you describe. Went on the Astra forum and various sensors were suggested, as was the fuel pump and fuel pump relay - trouble is, all cause much the same symptoms. In my case the problem was the fuel pump (I've now changed it and had no more problems) and it was very easy to identify that it was the culprit, just needed my ears. When you turn the ignition on on the Astra (last click on the ignition key before actually turning the engine over) the fuel pump primes and you get a very brief buzz/hum while it does this before shutting off again until the engine is running) - the hum is literally for 1-2 seconds, but is very noticeable when you know what to listen for with the rear seat base lifted or your ear to the fuel filler hole and an assistant turning the key. I'm guessing the same is probably true for the Omega - someone on here will probably confirm this or say otherwise if I'm wrong, but I'm sure I've heard the pump priming under the boot floor of my Omega estate. Wouldn't do any harm to try listening for this - if you don't get the hum the fuel pump or relay may be the problem.