Solved (STUPID mistake)
Please delete if pos.
How is the OLED powered and what voltage do the inputs require ?
You raised a good point which I have tested.
I successfully reconnected the display to my UNO and ran a simple test sketch I made with it connected to 5V on the Arduino, which worked as expected. I then connected it using 3V on the Arduino and reset the Arduino and, contrary to what I expected, it still worked perfectly.
From that I assume that the power is not an issue as the ESP8266 is providing 3.3V, but please correct me if I'm making a silly mistake. 