Generally speaking the Bosch VE injection pump used on tdi engines is very reliable and it would not tend to experience a sudden loss of injection pressure unless the cam plate is broken or all four delivery valves have somehow catastrophically broken up. There is ofcourse a pressure regulator fitted to the pump which can clog with dirt should debris find its way into the fuel / past your fuel filter but it is very unlikely and the engine would usually start after 3-4 turns either way.
The problem in your case sounds much more like air in the fuel caused by leak back. In other words, there may be a hole / fracture in a fuel line in the return side of the fuel system (injectors > injection pump > fuel tank). Split spill rail pipe work is usually the main suspect. If air can find its way into the fuel lines, fuel will inevitably run back to the tank and you will have to prime the system every time you start it, hence the reason why it is taking so long to fire up. Fuel will not necessarily find its way out of the hole where air is getting in.
You can perform a simple check to ensure that your lift pump is working correctly once the engine has fired up. With the engine running and maintaining a steady idle, slacken the 10mm bleed bolt on top of the fuel filter housing. You should find that diesel instantly starts to weep from the bleed screw. Do you see any bubbles in the diesel? If you can remove the bleed screw entirely without any sign of diesel emerging from the hole, your lift pump has likely given up the ghost or you have a severe fuel restriction in the fuel line (clogged filter / clogged pickup pipe / pickup pipe gauss) or potentially even a hole in the pickup pipe allowing air to be drawn into the system before the fuel pump).
Do you see white smoke from the exhaust when the engine is turning over and then for about the first ten seconds or so after the engine has fired up?
Be weary of Delphi lift pumps - they are not a quality item by any means. Unfortunately they have fallen victim to outsourcing and subpar manufacturing just like many other 'respectable' manufacturers, Bosch included. I have had five fail on me within the last two years, two of which ****ed diesel straight into the sump diluting my oil.
-Tom