The statutory or contractual notice period can be varied in a number of circumstances.
This occurs where an employee is dismissed without notice - summary dismissal - for gross misconduct. However, subject to statutory procedures, unless there is a proper investigation and an appeal hearing, an industrial tribunal/arbitrator might find that the dismissal was unfair.
Breach of contract
The employee can also terminate the contract of employment without notice if the employer has fundamentally breached the contract by their conduct.
Right to waiver
Employers and employees can both waive their right to notice, ie the employer and employee can agree to a shorter notice period. This must be by mutual agreement; and neither an employer nor employee can opt out of the minimum legal periods when forming a contract of employment.
Pay in lieu of notice
This will be a breach of contract unless the contract expressly provides for it or the employee is willing to accept pay in lieu of notice.
Minimum notice periods
The employment contract can be varied by agreement between the parties, but the statutory minimum notice periods will still apply.
An employee who has been given notice of dismissal can give counter-notice to leave on an earlier date than the one on which the employer's notice period ends. The minimum statutory notice that an employee must give is one week, but usually their contractual notice period will be longer than this. For the purposes of unfair dismissal legislation, the employee will still be treated as having been dismissed.
If an employee who has been given a redundancy notice wants to leave before their notice expires, eg to start a new job, they can ask the employer to agree an earlier termination date. If the employer agrees, they will still get their redundancy payment.
However, if the employer objects they may withdraw the original redundancy notice and refuse to give the employee a redundancy payment. The employee could apply to an industrial tribunal which will decide whether the employee should get all, part of, or none of the redundancy payment.