(2-part post):
You've not stated when the old MoT runs out.
To stay within the law, a car must be both 'roadworthy' and have a current MoT certificate (assuming, of course, that it's old enough to require one).
If a car fails an MoT test but there's still unexpired time left on the old MoT certificate, it's perfectly lawful (as far as the MoT regulations are concerned) to continue driving it until the old certificate expires. However, the fact that the car failed the MoT indicates that it probably no longer complies with the 'roadworthiness' requirements of the Construction & Use Regulations (or related legislation).
So, if there's still time left on the old MoT your son would be extremely unwise to drive it (after failing a new test), both on legal grounds and with regard to his own safety, if it failed because of, say, a problem with the brakes. However, if it only failed on, say, emissions he'd still be contravening the 'construction & use' laws (but not the MoT regulations) by driving it (as well as polluting the environment) but, unless the car's actually belching out loads of black smoke, the chances of him being prosecuted are effectively nil. (As stated above, the old MoT would continue to be valid).
Of course, if the MoT has already expired (or expires on Friday), Buildersmate's answer applies. There is no 'grace period'.