@brianlister , getting the RCM sent to DJTighe is excellent. The RCM is usually the problem when there is a no crank, no start situation. Most all of us carry a spare RCM in the glove compartment as the RCM is a common failing and by having a spare you won't get stranded as it is an easy quick replacement.

Checking the ground and positive battery cables is fundamental with a no start problem. You have recently replaced the battery so both battery terminal clamps should have been burnished and cleaned before reconnecting. The black negative wire from the battery goes to the fender near the battery. Disconnect the nut on the fender and clean the connection well. Same thing with the large red wire from the positive battery terminal down to the alternator. Just check and see that it is clean and without corrosion, but if it looks bad then turn on all the lights and the high beams, turn on the radio and the heater/A/C fan before disconnecting the negative battery terminal. Then the positive red battery wire and then then you can safely disconnect the wire down by the alternator and get it good and clean, before reconnecting. If you have gone that far get under the car on the drivers side and check the engine ground strap. It is located where the aft leg of the front wheels lower suspension arm connects to the frame. It is a copper colored braided round large wire about 7 inches long between the frame and the bell housing. Check it is clean and making good connection to the frame and the transmission. If you measure more than 0.5 ohms on any of those cables replace them. Reconnect the red positive battery cable, then the black battery cable and then turn off all the lights radio etc. By having all the lights on you help reduce shock to the electric modules that are connected all through the car.

Are any warning lights turned on in the instrument cluster??
Did you get a chance to check the starting voltages as shown in the Pulse Module post #6 ??
Does your FOB lock/unlock the doors and do the parking lights flash when you lock unlock with the FOB ??
You should check all the fuses to be certain electrical connections are indeed getting power; the small black plastic box in front of the battery, the fuses on top the RCM, the fuses in the engine bay back wall driver's side and even those in the driver's side of the instrument panel when you open the door. Most have nothing to do with starting, but some do and best to be sure.