Screeching from engine when starting jaguar xk8 4ltr

Don't mess with it and take round to yer local Jag mechanic right away before ya blow the hell out of it and you'll be up for thousands of $ instead of a coupla hundred.

get your diesel serviced.
or learn to find locations of noises.

Could be a belt screeching. Fan belt, A/C belt, Serpentine Belt, alternator belt, etc. Easiest way to isolate which belt is to open the hood and use your ears to see where the screeching is coming from. Left, right, front, behind the engine? Spray some WD-40 on the belt you think is making noise and if it quiets right away, then that is the belt making the noise. Either the tensioner needs adjusting or the belt needs replacing. The WD-40 does not hurt the car, it'll start screeching again in a few days when it wears off.

what does it sound like ,,, big high pitched screech is loose fanbelt ,,,,

maybe the same problem and same solution be done

Misalignment of the belt, incorrect belt, worn or loose tensioner, Too tight of an adjustment (many service centers do this on purpose so when the belt breaks in it becomes properly adjusted, I do not believe in this philosophy), The belt is actually one long belt called a serpentine belt.....

You have either worn out connecting rod bearings or a weak oil pump it is possible the oil pump pick up screen is clogged.

Fix your car right away, by clicking the link I attached
JAGUAR XK8 1996 2006 WORKSHOP REPAIR SERVICE MANUAL

See if it goes away when you turn the A/C off. The a/c comes on when you sre in the defrost so yo don't hit the windshield with hot air. The screaching is more than likely the A/C clutch trying to engage, that why it was overlooked or ignored by the mechanic. It is a common thing. The battery should be allowed to recoup from the starting of the car which a few minutes of running woll help so the a/c clutch gets full battery voltage to function.. Turning on the defrost and /or the A/c after about 3 minutes of warm up. Open your hood and have someone turn the a/c on and off, you can hear and sctually see the a/c clutch doing its irritating thing.But keep your fingers out off the engine and don't lean over it with loose clothing. Good Luck, Ned

Probably your fuel pump, have it looked at by a mechanic.