Following the district judge’s decision, either party can appeal to the High Court provided they lodge the appeal within seven days. This is an automatic right of appeal and the High Court must hear the case. When the appeal is heard is up to the court, but there is normally a wait of at least a few months.
One or two senior judges hear the appeal. Their decision ends domestic proceedings unless:
The High Court certifies there is a point of law of general public importance and
The Supreme Court grants leave to appeal
The appeal would then be heard by the Supreme Court. If the High Court refuses to certify a point of law then neither side can appeal against nor judicially review its decision. They also cannot ask the Supreme Court to consider an application for leave to appeal if a point of law has not been certified.
The parties have no further avenues of appeal within the UK if the High Court refuses to certify or, if it does certify and leave is granted, after the Supreme Court has made its decision. If the decision of the UK courts is not to extradite then there will be no extradition.
If the extradition is proceeding following the UK appeal process, the individual facing extradition can ask the European Court of Human Rights to consider if their rights are being infringed by the United Kingdom’s actions. Like the Supreme Court, leave to appeal must first be granted by the European Court. How long it takes to reach this decision or to hear the appeal is decided by the court.