Is it a crime to open other peoples mail in the UK?

Yes it is a crime to open other people's mail in the UK, this in interception of communications which is a criminal offence under the Regulation of Investigatory Powers Act 2000. This also applies to emails, phone calls and text messages.
However, the crime is only committed if the mail is in the course of transmission. Which basically means that if it has already arrived at your address then it is not interception if someone else just opens it.

If the person steals the contents or uses them to commit a fraud (for example) then they may commit other offences.

Some government agencies (such as the police) have the power to open mail even if it is in transmission, but only when authorised by a warrant signed by the Home Secretary who will want to know the reason why.

Customs Officers do not have to have permission, but can only open items entering or exiting the UK.

Sometimes the Post Office will open mail if it has been incorrectly addressed in order to work out where to deliver it or to return it. This is lawful because it is part of the agreement in posting a letter.