The National Museums-Liverpool Website has an article on passenger lists at https://www.liverpoolmuseums.org.uk/archivesheet31
It says, in part:
Britain made no attempt to record its emigrants or other passengers during the 19th century. The National Archives (PRO) holds some record of aliens arriving in British ports, from the late 18th century onwards amongst its Home Office records, eg HO 3 contains returns of alien passengers made under the Aliens Act, 1836, by masters for voyages made between 1836-1869 (except 1861-1866). Another possible source of information could be the passenger lists of HM steam packets carrying passengers to, from and within the Mediterranean area, 1831 to 1834 (ADM 30/35). (For further information consult the National Archives leaflet, Passenger Lists Domestic Records Information 56 available on line at www.nationalarchives.gov.uk).
Most records of passengers leaving the UK have not survived or are hidden amongst records of colonial departments. For example, registers of passengers travelling to New England, Barbados and other colonies for 1634-1649 and 1677, can be found in the Exchequer: King's Remembrance: Registers of Licences to pass beyond the seas (E 157). Passenger lists for America in the 1630s can be found in Colonial Office records (CO 1), which also includes registers of cabin passengers emigrating with the New Zealand Company, 1839-1850 (CO 208). (See the National Archives leaflet, Passenger Lists Domestic Records Information 56 available on line at www.nationalarchives.gov.uk).