I got bluetooth working last night on the clock really quickly. Steps are as follows.
1. Plug in bluetooth board to Arduino programming ports, 5V to 5V, GND to GND, RX to DOUT, and TX to DIN.
2. When the clock is powered on, the bluetooth board should light up and you should be able to pair with your computer.
3. Connect using a serial application like CoolTerm with the Baud Rate set to 115200 and new line character set to LF with local echo.
4. Type $$$ to enter command mode.
5. Set the name using:
SN,Nixie Clock
6. Set the Baud Rate to 9600 or whatever you got in the Arduino’s setup function, Serial.begin(9600);
7. Reboot the bluetooth device and repair with now named Nixie Clock bluetooth device.
8. Now you can connect with the Baud Rate set to 9600 and talk to the Arduino.

When I have completed the code, I will work on releasing all details about making this clock.