For liverymen and would be liverymen, the usual procedure is to become a freemen of your livery company first, and then apply for the City Freedom. [doing it the other way around brings no advantage, and can add small administrative complications]
All Freedom applications (other than those by Patrimony or Servitude) to the Chamberlain's Court are then approved by the Court of Aldermen. This link takes you to the timetable for their meetings, and hence the Chamberlain's Court timetable for submitting applications.
It is a pre-requisite to becoming a liveryman of any company that you must first have become a Freeman of the City. Hence the recommended timetable would be:
Freedom of the (livery) Company
Freedom of the City
Livery of the Company