Well clearly to go round the world you have to go through either Asia or Australia (or both if you wish). However, you have a problem in that Lusaka only appears to be connected to London in the OneWorld system, which means that you are going to have to use at least one sector by some other means to be able to use a OWE RTW ticket.

Note also that the UK is an expensive place to buy business class tickets - you could save a lot of money by getting a ticket starting somewhere else (e.g. GIB) and this might also help you with the routing to get you in and out of Africa.

The basic structure would then be:

GIB-LGW//LHR-LUN <overland to Tanzania/Uganda/Kenya> DAR/EBB/NBO-(x)LHR-EZE-<rest of S America> - <N America> - Asia - LHR//LGW-GIB

Where "overland" means either just that, or a separate ticket on another airline. You have 20 segments in total. You can't stopover in London en route from Africa to S America. Let us know where in S America, USA and Asia you'd like to go and we can fill in the details.