Right, tried a few more things. The difficulty is that flights leaving Cancun are different on different days of the week. I did find a flight UA1664, that departs today around 6pm direct to LAX, but I can't seem to book it on the UA website. It's on their timetable.

I tried an RTW itinerary over the phone with a travel agent in the UK. Got a price of £1340+490 taxes, which is a bit too high for me to get aid by work.

My current thinking is:
LHR-IAD-CUN on UA
CUN-LAX-NRT-TPE on UA/ANA
TPE-SIN-LHR on SQ - I want to try the A380 even though I'm in economy.

If I can find a way to get that CUN-LAX flight that should be the trip done.