Nobody is ever obliged to tell their employer about any convictions or cautions unless their contract (or a professional code of conduct imposed by a statutory regulatory body) says that must do so. However, many jobs involving contact with children or vulnerable adults do have such a condition in the contract.
An employee can't be disciplined or dismissed for an incident that happened outside of work, unless that action leads to a 'loss of trust' in the employee's ability to continue in their present post. (For example, if a teacher appeared in court for beating up his wife, the school's governors could legitimately sack him, because many parents might refuse to allow their children to attend his lessons. However, it might be harder for the governors to establish a 'loss of trust' if the teacher was before the court for a minor motoring offence).
Chris