HMRC have a very good webchat service which is what I use to contact HMRC. I have successfully used it a couple of times to get my tax code corrected on the spot.
Use it during UK business hours. Initially you will get a "digital" assistant, then you ask to speak to a human. You will go into a queue and you will eventually chat with a "real" person.
I've never had to wait very long and no call charges. You can also save the chat for your own record of the conversation.
https://www.tax.service.gov.uk/ask-hmrc/chat/online-services-helpdesk