Well, it is true that we are primarily a US-based legal forum but some of us have familiarity with UK and EU law such as the "Data Protection Act" and cookie laws since it's part of what you need to know if you're an Internet and technology lawyer!
The OP asks a very good question. Law should be similar both in the US and the UK as the principles are the same. I'll use U.S. law since that's what I know and you can infer what you wish from there. There are two issues that come into play generally and those are (a) copyright law and (b) contract law, e.g. the terms and use of the customer support environment. A third potential issue that usually concerns email: (c) privacy laws.
It's very possible that a customer may agree to certain provisions when they are provided support by a company, which includes that they are not given any license or right to publish private support communications to the party receiving support. So according to contract law (terms of use / terms of service) the user may not have had the right to make a post on a public forum. In addition to the breach of contract, copyright law may apply which protects any written content that would qualify as a "work of authorship" and receive protection. A short answer like "bleep you!" would probably not qualify as a work of authorship. But longer support emails certain could qualify. In the U.S., a
DMCA takedown notice would be appropriate in both circumstances to remove content from a third party website. Even though some might argue that some communications have an implied license to post content sent to another person, most agree that it would typically not extend to posting this information on the web publicly versus forwarding a post privately to another person.
I won't discuss privacy laws here which usually concern email communications and expectations of privacy.