Under s77 (fixed sum agreement) or s78 (running account credit such as credit cards and overdraft) of the Consumer Credit 1974 then unless they give you a copy of the executed agreement, they are unable to enforce any agreement. You should tell them what you have been advised and tell them that you’re not going to pay anything until they produce a copy of the agreement.
Can I clarify anything for you?
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You can all means ask for it but there is no provision in the Act for any money which has been paid over, to be returned. You can always argue that you paid this money by mistake (not knowing the statutory provision) and let the court judge decide. It would be interesting for the boot to be on the other foot for once.
Incidentally, under 77 and 78(1) you need to send them the fixed sum of £1 to get a copy of the agreement and then, it is sub section 4 a which says they cannot enforce the agreement.
Here is the statutory provision for you. http://www.legislation.gov.uk/ukpga/1974/39/contents