Superior courts in the hierarchy
At least in australia contempt procedures of a court or tribunal are spelled out in the legislation that establishes that court or tribunal. For example the NSW Local Court Act sets out the requirements for and maximum punishment for contempt in the Local Court and the NSW Industrial Relations Act sets out the same for the Industrial Relations Commission.
In all cases, contempt is an offence which may be criminal or civil but always requires the criminal standard of proof beyond reasonable doubt. Appellate jurisdiction can be carved out by the legislature but AFAIK it has not for any stripe of contempt. As such, the normal appeals process available for any other offence is open for contempt.