Tracing the wires from the living room wall to the boiler, I found a splice – important to note because it told me I was now looking for where the red and white wires connected to the boiler controls, instead of the red and yellow wires from the living room wall. This is truly an important but subtle distinction when you see where all this was leading:
With the power already off, it was a simple(ish) matter of unscrewing the terminals where the red and white wires (from the bundle on the top left) ran. Some of the terminal labels were easy to see: