It sounds like you might need to go to the newsgroup called
uk.legal
and post your question there.
I believe that even a verbal contract is valid but just harder to prove.
I suppose that a key question might be: Did you verbally agree to the contract they have contacted you about?
Another one might be: Did they send you the paperwork about this contract?
Some contracts have 7 or 14 day cancellation rights (I don't know if this is such a contract). If so, you may be able to argue that as they had not sent any paperwork, your right to cancel was denied to you.
Now don't hold me to this as no way am I a lawyer. It sounds to me as if they have handed it over to their own collections department for now and that these people have no legal way of extracting the money from you without some kind of court order.
It may be that they are threatening to inform credit reference agencies about your alleged non payment which could cause a hiccup with getting credit for a while into the future. If this is the case, you do have the right to access your credit file from credit agencies and to challenge and have altered any incorrect information.
It is not clear from your post who exactly they are handing the alleged debt over to.
I know more about cleaning windows than legal stuff which is why I'm a window cleaner and not a lawyer

I do believe that the CAB can put you in touch with lawyers who give you a free half hour consultation. I think you need to contact the insurance people (in writing) and get them to clarify, in writing, what exactly they think you owe them, why, and who exactly they are talking about handing the debt over to.
It may well be that they are hoping that your nerve won't hold and that you will pay up.
I've stretched myself with this post already. I think you certainly need more assistance than I (and probably anyone else) on this forum can give you.
The best I can do is say don't panic and get the full facts as far as you can (ALWAYS in writing).