My car will not crank over, new battery and starter

Your first two sentences suggest the battery was run down, most likely from a system draw. A different battery will solve that until it runs down too.

Here is a guide to help you see what to problem is:

https://www.2carpros.com/articles/car-battery-dead-overnight

In your third sentence, unless the new battery was also discharged, you likely have a loose or dirty battery cable connection. You can verify that by turning on the head lights and watching their brightness. Typically you will see they are fairly bright, then when you try to crank the engine, they get real dim or go out. You can quickly figure out which cable connection is causing the problem with a voltmeter, but if you do not have one, you may also see one of the cable clamps spark or smoke when a helper tries to crank the engine.

This guide can help as well

https://www.2carpros.com/articles/starter-not-working-repair

Once the cables are repaired, have your mechanic test the charging system. If you do have a voltmeter, measure the battery voltage with the engine running. It must be between 13.75 and 14.75 volts. If it is low, the battery will never fully charge. If the voltage is okay, the second half of the test requires a professional load tester to measure full-load output current and "ripple" voltage. If output current is one third of the alternator's maximum rated current, and ripple voltage is high, there is a failed diode in the alternator. 30 amps from the common 90 amp alternator is not enough to run the entire electrical system under all conditions. The battery will have to make up the difference until it slowly runs down.

Please let us know what happens.

Cheers

First we have to determine if the battery is being run down by a drain when everything is supposed to be off, or if it is not getting recharged by the alternator when the engine is running. Start by measuring the battery voltage with an inexpensive digital voltmeter. A good, fully-charged battery will measure 12.6 volts. A good, but run-down battery will read close to 12.2 volts. That one needs to be charged at a slow rate for an hour or two. If you find it near 11 volts or less, it has a shorted cell and must be replaced.

Once your battery is at 12.6 volts, there will be no problem cranking the engine if the cables and connections are in good shape. When the engine is running, measure the battery's voltage again. You must find between 13.75 and 14.75 volts. If the voltage stays around 12.6 volts or less, the alternator is not recharging the battery. At most you will get about an hour of driving time before the battery runs out, and less if the heater fan, head lights, radio, or other accessories are on. If the voltage is within that acceptable range, that just means the alternator is doing something. It still can have a problem that causes it to be able to deliver only one third of its rated current. That will also lead to a run-down battery, but it will take longer for that to occur.

Glad to hear you got it fixed. Before I left the dealership in 1999, so I deferred to a former student who works for a Chrysler dealer. There are two things to consider. The first is my standard test for a "voltage drop", which is just a different way of finding a bad connection. Watch the brightness of the headlights or interior lights when you try to crank the engine. If they get real dim or go out, there is a bad connection we have to find. That is where voltage drop tests come in to play.

If they do not dim, the starter system is not even trying to engage. That used to be commonly caused by a failing neutral safety switch, but today that is much more likely to be caused by a computer. It is standard practice now to use some type of memory saver device when replacing the battery so all the dozens of computers retain their memories. You may have some computers that need to be reprogrammed before they will allow the engine to crank. That is a trick GM came up with to make money from their unsuspecting customers, and unfortunately, most other manufacturers followed their lead a few years later. If it comes to that, I will have to get my student involved.

If the ASD relay goes it will shutdown the coil and injectors therefore no start.

Could be the starter, starter relay, transmission position switch and ignition switch, assuming the battery condition and connections are good.

How about neutral safety switch?

Is it cranking over and will not start or just will not crank over at all/nothing when key at cranking position?

Test the NSS as suggested by Doc

How was NSS tested?