I did think of simply trying to detect a voltage across the timer switch. This initially sounded simple and had the benefit that I wouldn't need a case as I could have hidden it all inside the machine but looking into it, I'd need an optoisolator that I don't already have and I realised that I'd have to have the ESP8266 outside the machine so it could pick up the wifi.
I'm hoping that I can demonstrate to my wife that all those boxes of electronic bits that she sees me playing with aren't entirely useless. Wish me luck.