Arduino UNO/Nano communication with SP1ML-868 module

Not an answer to my question per-se, but I was able to solve my problem.
As mentioned on my comment on the last answer, the SP1ML module's serial communication is handled through an ARM microcontroller. So I got my hands on an Arduino Due, which has an ARM Cortex M3 MCU, and uploaded a serial echo code similar to the one on my question, only now using Serial3 to talk to the module:

void setup(){
  Serial.begin(9600);    // Connected to PC
  Serial3.begin(115200); // Connected to SP1ML, Rx-Tx connection
}

void loop(){
  if (Serial.available() > 0){
    Serial3.write(Serial.read()); // Echo to SP1ML
  }
  if (Serial3.available() > 0){
    Serial.write(Serial3.read()); // Echo to serial monitor 
  }
}

I tried the exact same code on an Arduino Mega (same pins, ATmega2560 MCU) and it didn't work.
Obviously the MCUs are very different, but I have no idea one works and the other doesn't. A wild guess would be that the word sizes are different. In the Atmega, the word size is 16-bits, and in the ARM, the word size is 32-bits.

For some weird reason, this only works when the Arduino RX is connected to the module RX, and TX with TX.

This works because in that configuration you are actually using your Arduino as a USB-serial adapter. When you make the RX-RX, TX-TX connection you cannot communicate between the ATmega328P on your Uno but you are also making a RX-TX, TX-RX connection with the ATmega16U2 chip on your Uno that is used as the USB-serial adapter on the board.

I feel that it's problematic to try to use Serial (pins 0 and 1) for communication both with the computer and the module. Instead I would recommend using the SoftwareSerial library for communication between the Uno and the module.

SoftwareSerial doesn't work reliably at 115200 so you should first use the current working wiring to configure the module to communicate at a lower baud rate. 9600 is quite safe but if communications speed is a bottleneck for your application you can probably use a higher baud rate.

Next, wire the module to any unused pins on your Uno other than 0 and 1.

Finally, upload the following code to the Uno:

#include <SoftwareSerial.h>

SoftwareSerial SP1MLserial(10, 11); // RX, TX

void setup() {
  Serial.begin(115200);  // set this to the baud rate of the Serial Monitor
  SP1MLserial.begin(9600);  // set this to the baud rate of the module
}

void loop() {
  if (SP1MLserial.available()) {
    Serial.write(SP1MLserial.read());
  }
  if (Serial.available()) {
    SP1MLserial.write(Serial.read());
  }
}

This is the equivalent of the simple echo code you posted above and can be used to verify that everything is working before moving on to more complex code. I have arbitrarily chosen pin 10 for the Uno software serial RX pin 11 for the TX pin but you can modify the code to any pins that are convenient for you. You should make the connections with the module as RX-TX, TX-RX.