It is part of the pump but what makes you think it has packed up?
There is almost certainly a starting-up technique to follow which we tend to forget about, and as often as not this involves either FULL THROTTLE to start, or MORE COMMONLY, NO THROTTLE AT ALL.
The cold start devices alter the injection TIMING and sling in EXCESS FUEL but ONLY if two things:
1. the engine is turning very slowly, like at starter motor speed, and
2. there is no throttle pedal, i.e. the injection pump is looking at IDLE speed.
If your pump is like that and I think it will be, first make sure the glowplugs are in top class condition, and that they glow a decent time before you turn it over on the starter with the clutch pedal decked but NO THROTTLE. Just seeing the glow-light on the dash does NOT mean any of the plugs are actually glowing!
So, I reckon it's far more likely if the car runs fine, but is dismal to start is the glow plugs being a bit sluggish ... or perhaps one or two hardly working at all. These engine need a good glow to make them go!
As far as the pump goes, sling in a can of diesel cleaner when your fuel tank is very low, and take her for a drive. Sometimes works wonders.
CharlesY