Is it illegal to drive without an MOT?

Probably, yes, however it only becomes legal in the case if a car is on its way for an MOT test, that has been already taken an appointment for or also in the case if you are driving your car to or coming back from a workshop where it is getting overhauled. You can still be stopped by the Police, they might ask to see evidence of your appointment. 

Concludingly, as mentioned previously, if your vehicle is under three years old, or over 40 years old, you're allowed to drive it without an MOT.