Your erroneous level shifting attempt from Bluetooth TX to Arduino RX is the root of your problem.
Pins 0 and 1 are also connected to the on-board USB interface chip via a pair of 1kΩ resistors. These resistors serve two purposes:
- To "weaken" the drive of the USB chip's UART interface (actually only needed on the USB chip's TX -> Arduino RX connection), and
- To protect both the USB chip and the main MCU from damage should two connected pins be set to opposing levels, which would cause high currents to flow through both output drivers.
The TX pin of the USB chip is actively driven HIGH all the time when idle (stupid design by Arduino, it should be Hi-Z when the port is not open), and the RX pin is pulled up by the internal pullup resistor in the USB chip. Further, when the Arduino's
Serial is in use the TX pin of the Arduino is actively driven HIGH when idle (or LOW when transmitting a "1" bit).
The actual circuit you have looks more like this:
simulate this circuit – Schematic created using CircuitLab
Transmitting from the Arduino to the BT module is straight forward enough - the "high" point of the circuit is the Arduno's TX pin at 5V when idle. Current flows through R2 to the USB RX pin, and through R5 to the BT RX pin, with a portion split off to ground, which is what you want for your level shifting.
Communication in the other direction, however, is different, because you have two balanced "high" points in your circuit - one at the USB TX and one at the Bluetooth TX - and the fact you have 1kΩ resistors in both branches means that neither transmitter is "stronger" than the other. This means that when the BT TX is LOW (0V) you have around 2.5V at the Arduino RX pin (5V over two 1kΩ resistors = 2.5V in the center tap - see here).
Since 2.5V is neither within the Arduino's HIGH nor LOW voltage ranges (which are more than 3.5V for HIGH and less than 1.5V for LOW when running from 5V) the input is completely ignored.
The 3.3V transmitted by the Bluetooth module is a little under the official threshold for a HIGH signal on an Arduino, but in practice, it usually is detected fine. So remove the resistor network from the Bluetooth's TX pin and connect it directly to the Arduino. Since the input threshold voltage is a function of the supply voltage (0.7 * Vcc) it depends very much on your power supply as to whether it will work reliably or not. If you find it doesn't, you should use some upward level shifting. There are simple MOSFET based bi-directional level shifters available cheaply that are a better solution than resistors, and will allow proper operation in both directions, shifting a high voltage to a low one and a low voltage to a high one. These are good for lower frequency communications such as UART. If you should need higher frequency, or you find a simple MOSFET based shifter isn't working for you then a 74HCT08 or similar chip (be sure to use the HCT range not the HC range) or a dedicated level shifting chip may be in order.