Looking at ACSI website they give three methods to cancel a subscription. Mail, phone or online help/contact form, with a direct email link shown.
https://www.campingcard.co.uk/customerservice-and-contact/
Here’s the email address
shop@acsi.eu
The ACSI website gives the title of the email to be used too.
“ End your subscription - CCA 2020
Perhaps not surprising that replies to the phone aren’t good as no doubt coronavirus can be affecting things. Have you tried HMRC recently?
Having used their online contact form I’d very much doubt that any credit reference could be affected given that you would have evidence of cancellation, you’ve not paid anything further and the items are therefore unsolicited. I assume that they haven’t taken a further payment.