I'm at a loss,
After being AA'd home with what they diagnosed as a faulty fuel pump I spent Christmas carefully cutting the inspection hatch to change it (bought a new pump as well)
Get to the pump, disconnect the power socket and start cleaning mud off the tank.
Eventually think "I'll check the electrics" and 0V at the pump.
So I have tried connecting it to a separate battery and it works,
I stuck a multimeter in the connector for the pump, and 0V, I changed the relay at the other end and checked the continuity from one to the other, got a tone.
then I got 12V at the pump connectors yippee!
But the pump doesn't run... neither does the new pump I bought to replace that one.
I cleaned the connectors on the plug with both a tiny file (actually a drill bit) and alchohol and got some filth off.
The pump runs off the 12V from the spare battery (actually its 10.8V cos its flat but still runs the pump) but not from the big, main fully charged main one.
And I'm stumped,
Have also found oil in red connector on ECU but to my (limited) mind that would just mean no electricity to the pump, but there is electricity at the pump... (not enough current?)

Any ideas gratefully received

Assuming I understood you correctly after half a bottle of wine...
Continuity just means at least one strand of the wire is intact and working, or that it's not so degraded that it can't pass the test, but could still be in **** state and not able to pass enough current. You're best bypassing the oem cable and running a known good cable from the battery to the pump.
Had the same problem on my horn. Passed a cont test, but wire was ****. Bypassed it and all good.

I'd be having a good look at pump earth. It's not a lot of use having 12 at the +ve if the remains can't get to round.

Another thing that's fairly quick and easy to do is check that the power to the pump is switching on when it should. There's a wire from the ECU which becomes live when the ignition is switched on that trips one of the yellow relays under the driver's seat. This allows power to go to the pump. I think it's via a thick purple and white wire, but I'm not going to go out in the rain to check mine just now. Once you've identified the relay it can be worth pulling it out and bridging the load terminals and listening for the pump coming on. I find one of my partner Xylia's hair clips is ideal for such investigations.

Roadside mechanics invariably blame TD5 problems on fuel pumps. Yes, they can be a problem but not always. It's worth making your own checks before buying one!