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 Thursday, doing an overnight in Honolulu and then flying to Majuro on Friday morning. Most people recommend flying through the US, and then connecting to your flight to Honolulu.
Via Honolulu
12:20PM Friday Jan 10 – Depart London Heathrow to San Francisco
4:47PM Saturday – Depart San Francisco to Honolulu, arriving into Honolulu 720PM
Overnight in Honolulu
7:25AM Sunday – Depart Honolulu to Majuro – please see travel from Honolulu to see the remainder of the flight
Return
Monday Jan 20 – Check out of Beran Resort
7:40PM – Majuro to Honolulu flight
arrive into Honolulu 2AM Sunday Jan 20 (because of dateline you will arrive the same day you left Majuro)
7:10AM – Sunday Jan 20 – Honolulu to Los Angeles
arrive into Los Angeles at 230PM
550PM – Sunday Jan 20 – Los Angeles to Heathrow
arrive into Heathrow 12:20 PM on Monday Jan 21