Unknown CCJ to Old Address Leading to Mortgage Decline

There’s a misconception in this thread and others that you cannot dispute a default judgement if the creditor has been paid. You absolutely can and if anything shows you in a good light to the judge.

If your set aside application is based on your claim that the debt isn’t owed that’s one thing, but in an instance like this where there is a dispute regarding the process, you absolutely can set aside the claim.

I agree, there are misconceptions in this thread

To apply for a set aside, the OP would need to complete and file form N244, and pay the appropriate fee which is £255.
If it is not paid in full, then the application will not even be considered.

There is a possibility in some cases of a set aside by consent for the amount of £100
Where consent is obtained, yes it is essentially a rubber stamping exercise by the court. (although it is still the court, not the claimant, who decides whether or for to grant the set aside. Even with consent, a set aside application is not guaranteed to succeed.). This can only be achieved if both parties are legally represented (more cost!)
Where even one party is not legally respresented, the application must be reviewed by the judge.

However, in this case I understood the OP has already paid the sum claimed; the sum claimed not being in dispute.
Where the claimed sum has already been paid, the set aside application must always be reviewed by the judge.

And as I said before, set aside applications are normally only acceded to only where there is a realistic prospect of successfully defending the claim.

Moreover, why on earth would the claimant consent to a set aside in this case?
The usual reason consent is given by a claimant is because they want their money sooner rather than later (such as after the time and expense of a hearing), and certainly to avoid possible further costs and aggravation of seeking enforcement action.
So the claimant agrees to a set aside by consent, in exchange for immediate payment, rather than having to pay & wait for a hearing where the claimant may ultimately fail in their attempt ... or if they succeed, the defendant still may not pay up.

In this case, after what seems like many years (the judgement was entered Feb 2017, but it seems the OP moved out of the property at least 3-4yrs ago, so presumably returned the car before then) the claimant now has their money in full.
There is no reason for the claimant to co-operate with the defendant in this case in their request for a consent to set aside.

As you were invited to do, you can read up youself on 'a set aside by consent'.
Here's quite a good explanation:
https://www.masonbullock.co.uk/can-a-default-judgment-be-set-aside-by-consent/

Misleading advice:

Both parties do not have to have legal representation for a set aside by consent!

As long as the claimant agrees - and why wouldn't they in this case - the defendant paid as soon as possible after becoming aware!

(Though if the C won't agree then it's game over)

Misleading advice:

Set aside by consent does not require the C to admit making "some error"!

OP do your own research on this to clear up the confusion that the conflicting advice you are getting here must be causing you