Start by cancelling in writing, in whatever way the contract allows, and keep a copy with the date. If the company makes cancelling difficult or misleads you about your rights, that may already be an unfair commercial practice under existing law, regardless of when you signed up.
For the payments themselves, contact your bank or card provider. You can cancel a continuing payment authority or direct debit through your bank, and you can ask about a chargeback for payments taken after you cancelled.
For free help, contact Citizens Advice consumer service, which can also pass cases to Trading Standards. Some sectors have their own dispute schemes.
From 1 January 2027, breaches of the new information, reminder and cancellation duties also give you the right to end the contract and claim refunds. The Competition and Markets Authority can fine businesses up to 10% of global turnover for breaking consumer law, and has said it will act from day one against clear-cut breaches, although it expects to focus on helping businesses comply in the early months.