Coming from Europe you have a number of options, but we would recommend going through Honolulu.
This would be the most painless way to do it, arriving into Honolulu anytime on Saturday, doing an overnight in Honolulu and then flying to Majuro on Sunday morning. Most people recommend flying through the US, and then connecting to your flight to Honolulu.
12:20PM Saturday Jan 11 – Depart London Heathrow to San Francisco
4:47PM Saturday – Depart San Francisco to Honolulu, arriving into Honolulu 720PM
Overnight in Honolulu – we recommend the Modern Hotel
7:25AM Sunday – Depart Honolulu to Majuro – please see travel from Honolulu to see the remainder of the flight
Monday Jan 20 – Check out of Beran Resort – Fly to Majuro in afternoon
7:40PM – Monday Jan 20 – Majuro to Honolulu flight
arrive into Honolulu 2AM Monday Jan 20 (because of dateline you will arrive the same day you left Majuro)
7:10AM – Monday Jan 20 – Honolulu to Los Angeles
arrive into Los Angeles at 230PM
550PM – Monday Jan 20 – Los Angeles to Heathrow
arrive into Heathrow 12:20 PM on Tuesday Jan 21
This flight involves the “milk run” or the stop and go flight from Guam to Majuro, with 4 stops in between. This is 8 hours and a pretty tedious flight. We think flying through Honolulu is a better flight.