**UPDATE***
My car insurance with this company expires on the 6th June. Their customer portal says that it will auto-renew. I called up on the 15th May and asked for my No Claims Period proof sending - they sent this within 10mins of the end of the call. All good.
I also asked for the auto-renewal to be disabled on the account, as I've already sourced new insurance elsewhere.
Anyway fast forward to today, and I checked my credit card statement and lo-and-behold a £1488 payment for my renewalon the 30th May after I notified them to disable auto-renewal. The main reason I was switching is due to their renewal price being over double the price I found elsewhere (£690).
So I called them up today, explained it to their complaints team, and a couple of hours later I got a call back from Jenny at 1st Central Complaints Team - she had checked the call from the 15th May, and I did indeed ask for auto-renewal to be cancelled, and she confirmed the agent basically didn't do it. So she has refunded the policy in full and offered me £100 compensation which I will get in the form of a cheque.