Erm - what year? Really important...
The fuel pump isn't so much actually "priming" as it is developing system pressure. It is normal for the fuel pump to run, say, 3-10 seconds on pretty much any FI vehicle in order to develop system pressure - if you were to turn on the key and not start the vehicle, you'd hear the pump run for a few seconds and then stop.
If the screech is coming from the fuel pump, it will want replacing. Most (electric) fuel pumps are either "vane" type or "gerotor" type - but both screech when failing. A "vane" pump uses an vane wheel with springloaded "wiper" blades, and the constant volume change between the vanes is what develops both suction (fuel pickup) and pressure (fuel discharge.) Screeching in a vane pump usually means that one of the vanes has gotten hard, or has worn away to where one or more springs is contacting the inside of the champer.
A gerotor pump uses a set of nested gears, with one having one tooth less than the other. This results in another sort of "constant volume change" - and gerotor pumps usually fail due to advanced wear.
If you have a RENIX XJ (1987-1990 w/4.0,) the long start times are partly caused by the ECM itself - it's not that it's failing, it's just that the RENIX ECM does not start sending out ignition or fuel pulses until it A) reaches 300RPM and B) recieves a valid "sync" signal from the distributor (the RENIX will run without the Sync signal, but it prefers not to.) The fix for that, I have found, is to replace the start with a high-torque unit (which develops engine RPM faster) and make sure the alternator can keep the battery topped off. I have a local shop that knows precisely what I want when I come in, and makes sure I get it (I get the rewinds from him - I install them myself.)
Oh - make sure the screech isn't coming from the belt or the power steering pump - as they get old, Saginaw pumps can get noisy on cold starts - that's why I want you to make sure where the noise is coming from...
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