At this point you have definitely made some progress. After double/triple checking that you are using the correct pins on the HC05, I would suggest that you reassemble the first setup using the UNO and the USB. I mean the one that did work with the HCO5. If it still works, then you can have some confidence that the HC05 is still ok.

If the HC05 works on the Arduino UNO with USB. I have to wonder whether the 5V out pin on the Nano can supply enough current to the HC05 - it should, but I don't know and it may not be able to. In htat case, you may want to give the HC05 its own power supply with a regulator off of the 12V supply and the GNDs connected. But you may not need to go there yet.

Also, if you can post a clear picture of the set up with the Nano (and UNO also) it might help because a fresh set of eyes might see something you have missed.

Edited to add: Are you saying that the Nano setup works when you only use the USB to power the Nano and the HC05 and with no led/transistors attached - only the USB connection (no 12V), the Nano and the HC05 - does that power up the HC05?

The Nano setup works with USB power, as well as with USB and DC wall power - both or just the USB. It will not work with just the DC wall power (however it used to, so the Nano can - or used to be able to - output 5V from its 5V pin).

 

So, I remembered that I had purchased some BLE SPP boards a while ago. These were an HC-07 master/slave pair. I think that the slave is a comparable, but not identical to an HC-05/HC-06.

First I connected the HC-07 slave board to a cheap Nano clone to USB. Connections were simply +5V and GND and then TX to D10 and RX to D11. Next I downloaded and ran the simple program that you can see here https://tronixlabs.com.au/news/tuto...o-serial-wireless-uart-adaptors-with-arduino/

Next I installed this simple terminal app https://play.google.com/store/apps/details?id=Qwerty.BluetoothTerminal&hl=en to an android device.

I found and connected to the HC-07. When I send it a '1', an LED on the Nano turns on and when I send it a '2', it turns off. Works like a charm.


Next, I removed the USB and took the Nano setup to my bench and hooked it up to a bench power supply set at 12.0V. The connections were +12 to Vin, GND to GND. All the other connections (all four of them) remained unchanged.

I reconnected to the HC-07 using the Android terminal App and everything worked like it should (turning the LED off and on).


So, while it is an HC-07 and not an HC-05, everything works without a problem.

When you say, "The Nano setup works with USB power, as well as with USB and DC wall power - both or just the USB. It will not work with just the DC wall power (however it used to, so the Nano can - or used to be able to - output 5V from its 5V pin)."

I am wondering two things,

1. It seems like you had said pretty clearly that the Nano NEVER worked with the 12 V supply - that was the whole issue, but now I am reading that it did work at some time.

2. If your 12V supply or the HC-05 is damaged.

What works now?

Also, the board I used can handle 5V TX/RX, but the HC-05 that you were/are using may only take 3.3V TX/RX. That is a consideration.