Battery replaced and now car will not crank or start

Since the ignition switch sends a signal to the ecm through computer data lines, you may need to have trouble codes retrieved in order to find out what is going on but there are a few things you can do before that, first is check the "crank" fuse #5 in the underhood fuse block and if ok, try swapping in a horn relay or similar non important relay for the "crank" relay #34, also in that underhood fuse block.

If the check engine light is on, there must be a code stored.
Are you using a scan tool or just a code reader?
Even just a code reader though should read check engine codes, although if it is not connecting, that does not mean there are no codes.
Are you able to read the ECM/PCM module and if using a scan tool can you access other modules?

Without the battery charger hooked up, using just the key cranking the starter or at least trying to crank the starter does either battery cable smoke or get too hot to the touch?

Did you check the "crank" fuse #5 in the under hood fuse block and if okay, try swapping in a horn relay or similar non important relay for the "crank" relay #34, also in that under hood fuse block?