Regrettably I know of no on-line source of Lloyd's Lists. Original copies from the 1950s onwards are kept at the Borough Library in the City area of London which is a short walk from the Bank underground station in the City (Waterloo and City Line - train times were geared mainly to working office hours when I last visited). The Lloyd's List are bound in volumes containing a year's copies. Also available by prior arrangement are the old Lloyd's Voyage Cards for British ships (mostly 1950s-1960s era), and back copies of Lloyd's Shipping Index and Lloyd's Voyage Record. A great place to visit for research, but best to look at their website before planning a visit,
This of course is only of use if you can reach London easily.
Best Regards
Tom