Hi Saracenn

Welcome to Flyertalk. I've done a bit research for you, and I've got three "outside the box" solutions; prices quoted include all taxes and fuel, and based on weekday departure and 2-week duration. Add US$100 leeway for different connecting flights, weekends, etc. DO confirm your itinerary and prices with the airline's office. You can play around dates to get schedule and prices on ual.com and continental.com (or their UK sites for prices in pounds)

1. United

You buy three separate tickets:

LHR-IAD=LAX-LHR (open-jaw, essentially LHR-US return but from a different city) US$900

IAD-CUN-(ORD)-LAX US$800. Again US-CUN return with an open-jaw. ORD is for transit only as no non-stop UA flight from CUN to LAX.

LAX-(NRT)-TPE roundtrip US$1250.

Total cost: US$3000 approx.

Pros:
Star Alliance miles
Be Silver with Diamond Club/Star Alliance during the journey.
Plenty of United discount certificates around to save some $$ (I can help)

Cons:
Worse service and amenity in-flight than Continental's
More expensive than CO
Less frequency to and from CUN than CO

2. Continental

Again three separate tickets:

LHR (or Gatwick if you prefer)-EWR=LAX-LHR US$700
EWR-CUN-(IAH)-LAX US$750
LAX-(NRT/ICN)-TPE roundtrip US$1200 This leg requires you to fly one of CO's partners: BR (non-stop direct), KE (via Seoul) or Northwest (via Tokyo and another US city I think)

Total cost: US$2700 approx.

Pros:
Better service (CO is one, if not THE, best US legacy carriers)
Better schedule and frequency to and from CUN
More direct flight (via Houston) than UA
5000 miles for enrolment in Onepass, for Brits ^:
https://www.continental.com/web/en-G...5849D462A85D49
You'll be Skyteam Silver after the trip, and maybe Star Alliance Silver sometime next year as Continental is switching from Skyteam to Star Alliance

Cons:
Partners flight may earn less or no miles for their cheapest fares. Check Onepass website for which fare classes on which airlines can earn how many miles.

3. Skyteam RTW

LHR-EWR/ATL-CUN-LAX-NRT-TPE
then fly to any combinations of cities west/Europe bound within the rules.
Here's a link for reference. You should get quotes from multiple TAs:
http://www.westernair.co.uk/rtwskyteam.html

Pros:
The most direct flight path, can visit other places along the way
Free to change dates (I think)
Lots of airlines and routes to choose from
Can put all the miles in Continental's Onepass (as the CO option above)

Cons:
Expensive for miles per dollar
Carry less luggage (from Asia to UK, compared to from US)

You can save 1%-4% on your airfares booked online via eBates (www.eBates.com) portal. You reach the airlines/online TAs (Travelocity, Expedia, etc) via eBates, they get a certain % of commission for your traffic/purchase and you get 1% or whatever in return. If you are not a member yet I would appreciate you join with my referal (I can PM you) at no cost and you also earn US$5.

Cheers