Fuel pump comes on but no fuel pressure on mercury sable 2001 3.0 DOHC

At around 3:00 he will remove the vacuum hose from the fuel pressure regulator. https://www.youtube.com/watch?v=r8zY1AFEjt8

You are right that it could be either component, or something else electrical. The security system will kill the fuel system if the wrong key is used or the system is acting up.
A mechanic would check base fuel pressure with a gauge and also see if the pump is getting power from the relay when it fails to start.

Are you getting fuel pressure? You have an inertia switch in the trunk and it might need reset(push the red button down). It shuts off the electric fuel pump in case you have an accident or get bumped hard.Sometimes they just actuate for no known reason.
Also crankshaft position sensor could be dead. Have you run the engine codes? don-ohio (:^)

Run the codes,change spark plugs,clean the MAF sensor and always make sure to replace a fuel filter every 30K miles. don-ohio

Was the problem there before you replaced the pump ?
The one way valve in the pump could be letting fuel pressure drop to zero. The fuel pump relay may be faulty too.
Have you checked the fuel pressure before and after cranking the engine ?

change cluster 4 fuel gauge

i think with out book its in the tank

Check the fuel pump shut off switch.Its usally mounted in the trunk its for in case someone rearends you it will shut the fuel pump off.

Turn the key between the "off" and "run" position without engaging the starter. Can you hear the fuel pump turning on and off every time? The car SHOULD build proper fuel pressure, then shut the pump off until its been cranked or has sat for a while. If the pump never shuts off or is pumping every time its switched on and off, I'd say you've got a pressure issue or leak in the system. Testing fuel pressure isn't too difficult either. The tools can be "borrowed" from autozone or similar store if you don't own them already. Just buy them, do the job and return them clean and complete. I've never had a problem, even with "non-returnable once used" tools! : ) These stores can usually supply you with the proper specs if you need them. You may also want toi try testing the signal to the injectors themselves. Ahgain, tools avail at autozone and the likes. Good luck!

Check/ or replace the fuel filter first .Fuel pressure should be 40-50 psi and hold steady for 30 seconds after the engine is off.