There is no third-party right of appeal to the Planning Inspectorate if a planning permission is granted.
However, if a third party considers that the planning permission was granted unlawfully, and due procedure was not followed they can challenge the decision through a judicial review in court. The judicial review process can only consider the lawfulness of the procedure that the Local Planning Authority took in coming to the decision to grant planning permission, it is not to consider the planning merits of the case.
A judicial review has to be logged within 6 weeks of a date where a challengeable ground arises; this is generally the decision notice date for a planning decision but is not always the case. Anyone considering a judicial review would be advised to seek independent legal advice.