Where under any Act a place has ceased or ceases to be part of a borough or the liberties thereof, all matters by virtue of a local Act of Parliament or otherwise cognisable by a justice or by the quarter sessions having jurisdiction within that place shall be cognisable by the justices or the quarter sessions of the county, liberty, or jurisdiction within which the place is situate, in the same manner and subject to the same provisions as they were within the jurisdiction of the justices or the quarter sessions for that place.