Tehnically you are bound if you signed a contracct. But you can ask for a copy of the original.
In law, you don't get a cooling off period for a business transaction.
Therefore you are bound.
But the fact it was not signed on the dotted line, does not really matter - if you signed the contract, which you accept that you did, then it is binding,
If you refuse to pay then they can try and take you (not your wife) to Court.
If you do not live in England then the reality is, it will be hard to take you to Court.
You could dispute it and have it transferred to your home Court.
So whilst technically you are liable, enforcing the contract is an issue for the scam company.
Can I clarify anything for you about this today, please?
If not, I would appreciate a 5-star rating for my answer. If you need anything further I am available for a follow up at no extra cost.