Well, it's on a government website, so they OUGHT to know! However, I think they're playing safe and using some "weasel words".
The actual words on the website are:
"
Driving a vehicle that’s failed
You must not drive the vehicle on the road if it fails the test, even if the MOT hasn’t run out, except to:
- have the failed defects fixed
- a pre-arranged MOT test appointment
You can be fined up to £2,500, be banned from driving and get 3 penalty points for driving a vehicle in a dangerous condition."
So it doesn't say that driving a car with a failed MOT before the old one has expired is illegal, it says that driving a car in a dangerous condition is illegal. We kind-of all know that though! It's illegal to drive a car that doesn't comply with the Construction and use Regulations and the Road Vehicle lighting Regulations. The contents of both of those are largely what the MOT is based on. So, for example, if I fail an MOT because I've got a number plate light out, that's technically a fail, and contravenes the Lighting Regs so it shouldn't be on a public road. However, the same also applies if it HASN'T failed an MOT! If, for example, I got my MOT a month ago and my number plate light fails, I'm just as illegal as I would be if it had failed an MOT as a result of the number plate light.
Given that pretty much ANYTHING you could fail an MOT on would be a contravention of those regulations, I can't see how a government website could say anything OTHER than "you can't drive it" - even though, it's the non-compliance with the regs that is the illegal thing, NOT the "driving a vehicle with an unexpired MOT that has just failed the test" thing. On a government website like that, they'd never be able to go into a full list of things that could make a car "dangerous" as opposed to a list of thing that make it "techncially non-compliant", so the easiest thing to do is to just say "you can't".