As has already been mentioned, anytime you have to visit Africa outside of JNB you're going to find major problems making a complex RTW work. Seems your cheapest option would be:

1) RT LAX-LON which unfortunately won't be cheap Jun-Dec. I found $858 on ITA on BA.

2) LONE4 from LHR for about 1,400 Pounds. Something like:

lhr-cai-bcn-sjo-lax-sju-dfw-anc-dfw-eze-ccs-scl-ipc-scl-jfk-nrt-del-nrt-cgk-nrt-lhr

will get you the major destinations plus is about 72,000 miles for a well optimized MR. Might as well buy the LONE4 for 200 Pounds more and include South America for more miles and interesting destinations.

3) You can buy a seperate RT CAI-EBB for $544 in June on Egypt Air.

Total is about $4,200 not including tax on the RTW and will earn you a boatload of miles between the RTW and two seperate RTs. This also gets you home to LAX for a break partway through your trip at no extra cost.