I would find a "classic friendly" MOT test station and get it booked in, there's lots of testers out there now who don't understand older cars and the limits applied to them for testing (emissions rules are a good one)
You can take it to a test without MOT or Tax but MUST have insurance in place and MUST be booked in advance, when you book it in make sure they take your registration number and name and explain the situation to them in case you get stopped on the way there (it does happen in these days of ANPR) and take your insurance certificate with you.
If it passes, well done, Is your car registered to you at your address, do you have a smart phone? If yes to both, use it to get online and buy a tax disc there and then. Keep the confirmation email available on your phone in case you get stopped on the way home, then don't use the car again until the tax disc arrives, usually in a couple of working days but they say no more than five working days.
If no smart phone then get yourself to the nearest post office, ideally walk there, or if you can't then park right outside it and if you get any problems you'll arrive back to the car with the new tax disc and mot certificate in your hand and should be ok.
If it fails the test...copied from direct gov...
If your vehicle fails the test
You’ll get a ‘notification of failure’ from the test centre if your vehicle fails the test. The failure will be recorded in the secure central MOT database.
You can still drive your vehicle if it fails the test and its existing MOT certificate is still valid (ie you got it tested before the expiry date). However, you might be stopped by police and prosecuted if your vehicle is unroadworthy.
If the vehicle fails the test and the certificate has expired, you can only drive it to:
- a pre-arranged appointment at a garage to have the repairs done
- a pre-arranged MOT test appointment
Your vehicle should be retested at the same test centre which did the original test.
The reason for the last line is so the test station can carry out a partial MOT test (although only within ten working days of the failure date) by just checking the original failure items have been repaired correctly, but they will also check associated items too so ensure it has been repaired properly, I've lost count of the vehicles I have retested and, for example, found the brakes have been repaired brilliantly but the wheel nuts are left loose!
If you're unsure what is required to pass the retest then don't be afraid to ask, this is another common reason for retest failures, inadequate repairs due to lack of communication.
I've often sent people away without logging the partial test on to the system if they have time to put things right and get back to me before they run out of the "ten working days partial retest" rule, but believe me there's an awful lot out there who won't!
No good tester likes to fail a car presented for retest, If it's something simple to put right, like my loose wheel nuts example above, then a good tester will advise you, put it right, then send you on your way with pass certificate in your hand in the hope that you remember them next year for having looked after you!
Remember though that you only get ONE partial retest and a partial retest can only be carried out at the test station that failed the car originally but doesn't have to be by the same tester oddly, if you fail this then you'll be paying for another full test again. We can't get by this as testers because we have to record each test/partial test and the system won't let you log a second partial test, meaning you have to record all the details and measurements (brakes,emissions etc) needed for a full test again!
Hope this helps.