Firstly, a polite request: Please include some punctuation in your posts. I found that really hard to follow. Many people, when faced with a couple of hundred words without any punctuation, wouldn't even bother trying to read it at all.
You ask about your 'rights' but I think that 'reality' is possibly more important here.
Since what you've received (i.e. a car without tax) isn't what you were promised (i.e. a car with tax) it's likely that a court would rule that since your acquisition wasn't 'as described' you do have the right to demand a return to the status quo before the deal took place. (i.e. you return his car and he returns yours).
However it seems that such an exchange is unlikely to happen, since he doesn't even have your old car any longer. He would have to try to reverse the subsequent swap which (even though he might not legally have been in a position to make such a swap) he would probably find almost impossible to do.
Face reality and forget about trying to get your old car back. Instead, pursue him for the cost of taxing the car that you've received from him.
Write a formal letter to the guy as follows:
Put your name and address at the top right.
Put his name and address (lower down) near the top left
Put the date underneath that.
Then proceed as follows:
"DEMAND FOR PAYMENT
TAKE NOTICE that I have not received a Road Fund Licence disc for vehicle registration number .......... , in accordance with our agreement to exchange motor vehicles made on ..........
I therefore DEMAND payment of the sum of £........(that sum being the cost of taxing the said vehicle for a period of .........).
Further, TAKE NOTICE that if payment of the above sum is not received within 14 days I will instigate COURT ACTION to recover that sum from you,together with the COSTS of such action"
Add your signature and print your name underneath it.
Retain a copy of the letter and send the original by Recorded Delivery to the guy who you exchanged cars with. Go online, using the tracking number on your Post Office receipt, to note the date and time when the letter was delivered.
If you don't get your money, initiate court action here:
https://www.moneyclaim.gov.uk/web/mcol/welcome
Chris