To drive to the MoT with no existing MoT and/or tax it must be pre-booked -You must have insurance. This is valid for that journey, and the return, even if you fail.
To tax the car you must have insurance, MoT and LogBook (or reminder) or green slip if you have only had the car a short time - You can still be prosecuted for no tax even if you are driving to the Post Office to get it.
If you apply for tax online you are covered straightaway, although the only grace period is five days from the expiry of the old disc, or SORN, if you have applied for the new one.
If there has been a long period since it was taxed you can still be prosecuted for failure to display, but it'd take a very bored bobby to go to the bother.
Any vehicle used, OR KEPT, on a public road must be insured at all times. If there is no insurance then you must SORN the vehicle AND return the tax disc, even if it has NIL value.
You cannot keep a tax disc on a SORNed vehicle even if you intend to use it a few months later or it is only a summer car.
If you have no tax, or no insurance, or no MoT the police can seize your car if you drive it.
No MoT - Fine of up to £1000, this is non endorceable (no points)
No Tax - Fine of up to £2500, also no endorceable
No Insurance - Fine of up to £5000, 6-8 points and a discretionary disqualification.
There is more but that covers some of the new legislation