With regard to personnel, the strength of the Royal Navy in March 1965 was about 101,000 with a requirement for 102,500. The target for 1968 was up to 107,500 (Hansard link). With regard to vessels, Wikipedia shows numbers for 1960 and 1965 (link). For more detailed information Jane's Fighting Ships, published annually, shows the strength of the Royal Navy in terms of personnel and vessels.
(Incidentally, the Hansard link is to an account of the annual Commons debate on Navy Vote A, defined as "â€¦ the maximum number of officers, ratings and Royal Marines who may be maintained for naval service â€¦". The House was being asked to approve an increase to 104,000 and it makes for interesting reading.)
The National Archives at Kew also hold returns required by Parliament, dating back to the statistics obsessed Victorians. Data may also found in Parliamentary papers on defence, etc. You can search PIMS or contact the Parliamentary Archives.
Haven't done a 'Watch and Station Bill' since 1969. :? I'll need more than half a dozen of us though. 'action stations, defense stations, cruising stations, watch, part of ship, cleaning stations..... it goes on and on. Who would be a poor put upon CPOGI? :roll: We could just about man a whaler, under oars of course!