The wiring diagram shows that the wires are as follows:
Black is ground and there are 2 black wires.
Brown/yellow stripe is Power
Blue/ with white stripe is not used for the XK8.
You should get power between the brown and black when the key is turned on for 3 to 5 seconds to charge the rack. Then it will shut off power.
Then when the engine is cranking the ignition is ignored and the ECU takes over and provides power to the brown.
The initial is working so when cranking over does the power show on the brown to black? If it doesn't show power then the crankshaft position indicator is not telling the ECU that the engine is rotating.

First you need to be sure you are getting fuel. Turn the ignition on for 5 seconds and then turn it off. Go under the hood and on the right front side of the engine is a valve that looks like a tire valve with a plastic cap. Remove the cap and press the schrader valve with a rag covering it. If you get a blast of fuel then the pump is working. If no fuel then the pump is bad. The rack pressure should be 45 to 55 psi at that valve.
Check that out and get back to us with the results.
Also, it would be most helpful if you added what year model and engine of XK8 you are working on. The info is based on the 98 XK8 diagrams.

 

Wiring diagrams for your model are here at the top of the page . Is the car normally aspirated or supercharged that has 2 pumps to troubleshoot with .

http://www.jagrepair.com/JaguarXK8El...OBDIIcodes.htm

 

 

You can swap the relay with one of the same part # as a pitted relay power contact can show voltage and not have the contact area to carry the current .

You can look in the connector and see if you have burnt sockets .

look to be sure you have good ground wires . Location ?

Verify the relay clicks 3 times during starting sequence . On - Off - On .

 

Since you can read voltage at the pump when the key is energized your relay is fine. Most likely you are correct it is time for a new pump. The fuel line disconnect tools are very cheap just plastic rings with a cutout to go around the fuel line. Harbor Freight has a set of FUEL LINE DISCONNECT tools. If I remember right the white one is the one that works. You place the tool on the line with the smallest outside diameter facing the fitting and push toward the fitting. Then pull the fuel line out. Pushing in releases the holding ring. To reinstall just insert the line and push in.
The location of the lines was not one of Jaguar Engineering Department's greatest achievements.

 

Welcome to the forum Staffan,

I can see you are getting good advice on the fuel pump.

When you get a minute, please follow this link New Member Area - Intro a MUST - Jaguar Forums - Jaguar Enthusiasts Forum to the New Member Area - Intro a MUST forum and post some information about yourself and your vehicle for all members to see. In return you'll get a proper welcome and some useful advice about posting to the forum.

Graham