In January 1962 schedules, looks like the best time you can make is
Leave Honolulu on United at 2355 Thursday; arrive San Francisco in the morning and board a Pan Am 707 that arrives London 0705 Saturday morning; leave London at 1800 on a BOAC Comet and arrive Nairobi at 0850 Sunday.
Or you can do the same schedule but leaving Honolulu Saturday instead of Thursday.
One alternative is a BOAC or QANTAS 707 Honolulu to London, then the Nairobi flight after a 20-hour stop in London.