You need to give a better description of what you want to achieve.
The only tip I can give now is that on the Nano, RX and TX are shared with the USB communication. You can look at SoftwareSerial or (the better) alternatives NeoSWSerial and AltSoftSerial if you need a second serial port functionality in the Nano's software.
+-----------+ +-----------+
| | | |
| N TX |-------| RX c |
USB --| a RX |-------| TX o |-- USB
| n | | n |
| o | | v |
| | | |
+-----------+ +-----------+
Like this?
Should work (to my knowledge) if you keep the Nano permanently in reset.
First, there was a mistake in the drawing in reply #3. It should be TX to TX and RX to RX.
Explanation below
Nano Conv
+--------------+ +--------------+
| | | |
| +------+ | | +------+ |
USB --|---| FTDI | | | | FTDI |---|-- USB
| | | | | | | |
| | R T | | | | R T | |
| | x x | | | | x x | |
| +------+ | | +------+ |
| | | | RX | | | |
| | +-----|----->-----|-----+ | |
| | | TX | | |
| +--------|-----<-----|--------+ |
| | | |
This shows how the two TTL-to-USB chips (one in Nano, one in the converter) should be wired to be able to communicate. At the Nano side, I've left the actual pins out. I named the chips FTDI.
So the Tx of the one FTDI chip goes to the Rx of the other one and vice versa.
I've now also named the physical pins on the Nano.
As (hopefully) can be seen, the Tx of the 328P on the Nano goes to the Rx of the FTDI on the Nano. But that same Tx pin of the 328P is also the physical TX output pin of the Nano; same for the RX. So the Nano's TX pin needs to be connected to converters TX pin and the Nano's RX pin to the converters RX pin.
TX pins are outputs and one should never connect outputs together (unless they are open collector / open drain). The Nano has resistors for protection to prevent you from blowing the 328P's blowing up if you accidentaly connect the TX pins; indicated by R in above drawing.
====
So you have now two outputs driving against each other. If the 328P has its uart active, the 328P's Tx pin will be actively driven HIGH (serial idle state). If the converter's TX pin is now driven LOW, the FTDI on the nano will not see it because the 328P's Tx pin is HIGH.
If you keep the 328P on the Nano in reset, all 328P IO pins will be high impedance and the 328P will not have 'the power' to drive its Tx pin HIGH.