Hooking up the Numitrons to the TLC5940 went pretty much in the same way as described on the library's page:

The Numitrons are common anode so they work by hooking the anode to VCC and the filament pins to correct TLC5940 outputs. The Numitrons are listed as having an operating current of 17-22 mA so I used a 2 kohm resistor on the IREF pin. For some reason this lead to the Numitron filaments having very inconsistent glows. I changed the resistors to 1 kohm and after that the filaments had the same intensity.

The 1 kohm IREF resistor means that each output could potentially sink about 40 mA but the TLCs didn't seem to be heating up so I'm going with that setup.

The ESP8266 is a temperamental little thing but I've had good experiences using the Espduino library so that's used here for the MQTT functionality.

As the ESP's logic level is 3v3 I used a zener diode level shifter in front of its RX pin to bring the Arduino's 5V down to a safe level. The TX pin is used as it is.

The Arduino can't properly power the ESP so it's important to supply power to it independently.

The Tiny RTC module is I2C compatible so using it was pretty straightforward wih the Hobby Components library.