Will not start after trickle charging the battery all night with it connected incorrectly?

Hello, does the Check Engine light come on when you turn the key to the "on" or Crank position? It should come on for a few seconds. If not, then the engine computer is not being powered up. What size engine does your vehicle have and is it a Sedan model?

Okay, I have some diagrams here with some fuses to check, unfortunately there are no fuses on the ground side, so most of the time when this happens there is a module that is damaged. The Totally integrated power module (TIPM) is on the driver side engine compartment, it contains a lot of fuses and some relays, but it also has internal circuit boards. The main fuses that power up the engine computer are in the TIPM. Here are a few diagrams to start with, the fuses to check first are highlighted in yellow. I think there is one more page ill post that has a fuse and some other power feeds to the engine computer (ECM). If there are no burned fuses, then power to the ECM will need to be checked, I'm not sure if you have a basic multimeter or not, but hopefully the only module that took the majority of the current flow was the TIPM, and it might need to be replaced. Even with a used one.
Almost all the fused highlighted are for the ECM and the ignition node(module). So, check all these first and see what you find. Also take note of any connectors that look discolored or melted. I assume that charger was hooked up all night.

https://www.2carpros.com/articles/how-to-check-a-car-fuse

https://www.2carpros.com/articles/how-to-use-a-voltmeter

The red positive lead coming directly from the battery? This didn't damage the charger either.

Is there any power across the battery itself?

I just wondered if the battery ended up being shorted out due to being on the charger for so long hooked up backwards. I don't see any fuse in between the jump post and the battery.
If you have a multimeter, you can still go through and check the fuses with the meter set on dc volts, it should just read 0 volts, so 0 voltage drop across the fuse.

Not voltage at all, what if you disconnect the battery cables, does the battery read any voltage then? Sounds like the battery got fried as well.

Did the battery push any acid out of the top into the battery tray? I would go buy a basic multimeter that can read 20vdc, so you can test the fuses before hooking up a new battery. Most likely that battery is no good. But you can check and replace any fuses that are blown first, and then see if the battery will take a charge afterwards.
This fusible link will need to be checked as well. I'm hoping there is a high amperage fuse somewhere that blew first, I'm just not seeing one yet in the diagrams. The fusible link coming from the Alternator is probably the highest rated fuse.

With a multimeter you'll be able to check the fuses with the battery disconnected. Since the battery is obviously shorted out. You didn't find any fuses blown yet?

Okay, I'm sure you will find some blown fuses, and that fusible link may be down near the start motor. Sometimes they put them on the starter motor's main power stud. It will be a section of wire that is a different gauge than the rest of the wire, and most times they burn inside the shielding, so if there is no continuity from the Alternator to the battery positive, it's probably that fusible link, you can see it should change wire color before getting to the starter.
Also I would keep away from using the charger anymore, it might be damaged as well and you dont want it to catch fire. Im actually surprised it didnt being on there all night, I know batteries can explode from being hooked up to a charger and left like that.