A few years ago a school teacher in the US lost her job when it was discovered that she had previously performed in pornography. There was no suggestion that she was a poor teacher, it was simply the fact that she had an embarrassing past that got her fired. She had not mentioned it when applying for the job, it was only discovered by her students recognizing her in a video seen on the internet.
More recently the EU Right to be Forgotten has been used to prevent a web search associating innocent people with criminals mentioned in news stories.
The owner of Oink's Pink Palace was fired when he was arrested for copyright infringement, but subsequently found innocent. In such cases it seems that the accused often does not get their job back, despite being dismissed over something that turned out to be perfectly legal.
Generally speaking, in the UK what employment rights does a person have when their past is potentially embarrassing? Former jobs, former criminal convictions now spent, former involvement with "undesirables" etc. Can they be fired, and if not what specific laws protect them and to what extent?