If your employer takes disciplinary action against you, they’re legally obliged to tell you about their appeals procedures. This could be stated in their formal disciplinary letter to you, or it might be included in an employee handbook. If you can’t find any mention of an appeals procedure, ask your employer as soon as possible.
You should appeal in writing as quickly as you can, setting out the reasons why you believe the decision was wrong. There will then be an appeal meeting scheduled, at which you may be accompanied by a colleague or trade union representative if you wish.
If you remain unhappy with the outcome of the appeal process you can then bring a claim to an employment tribunal. However, this must be done within three months of the original decision.