Uploading to the Uno happens via the serial (UART) bootloader on the D0 + D1 pins. The BC68 shield also communicates with the Uno via UART and the exact D0 + D1 pins. So you have a pin conflict – when you want to upload a new firmware to the Uno, the switch must be flipped to the ‘UART to USB’ position so that the BC68 module does not occupy D0 + D1, but in order for the Uno to then be able to communicate with the BC68 at all, it must be flipped back to the ‘UART to MCU’ position after uploading.
That’s why the recommended development board in the user manual linked above is exactly not an Arduino Uno but an ST Nucleo L476RG board. An uno very unsuited for this since you by default only have 1 serial but the wanted usages is programming new firmware + debug outout + communicating with the BC68 module, which ideally happens over three different serial connections (while the first one may also happen in an entirely different way). With a Nucleo board (any Nucleo board, really), the programming happens via SWD (completely different pins and method as opposed to UART), the default
Serial is not on D0 + D1, meaning it does not by-default conflict with the BC68 communication UART, and it has tons (as in, 5)
Serial peripherals available, one also on D0 + D1, where the module can connect.
I would heavily suggest that you either exchange your Uno for any ST Nucleo board (e.g., F103RB, L47RG, …) as that gives you a much better experience regarding the serial connections and uploading and out-of-the-box working debugging. If it has to be an Uno, the best you can do is using a SoftwareSerial to implement the debug output. (You must use the hardware UART on D0 + D1 for the module communication as that is what the shield is hardwired for).
There are two more tests that you can do:
Direct connection: Unplug the BC68 shield from the Uno, flip the switch to “Main MCU to USB” and plug in the right USB cable into the USB port which is located on the shield. It should power it on and you should be able to start a serial monitor at 9600 baud and line-ending = CR+NL and type in AT commands, like
ATI, directly to verify that the module is alive. No Uno needed at all for this.
You can turn the Uno into serial-pass through device that accepts serial input (e.g. from an extra USB-UART adapter and a SoftwareSerial) and passes that through to the hardware serial connection on D0 + D1 (where the BC68 is connected to when the shield is plugged in). See this sketch. This would allow you to send AT commands to the Uno via the external USB-UART adapter which in turn outputs it on D0+D1 to the module, which hopefully responds, the Uno can read in that answer and re-output it on the software serial.