Firstly - I can't count - your original itinerary had 19 segments (only three in SWP) so you could have added LHR-ARN at the end a thus have a valid intinerary (but note that you'd then have 5 segments in Europe - allowed but at an extra cost) - Seems I'm wrong here - see post by wasabi tofu
With regards to starting in Africa, your suggestion of substituting CPT for ARN works at the beginning, but not at the end of the intinerary:
CPT-LHR(t)-DXB-AMM-MAD(t)-GIG-SCL-USH-EZE-SYD(t)-CHC-AKL-SYD-PEK-HKG-DPS-KIX-BKK-LHR-CPT
The intinerary involves going to Europe twice - this is allowed when one is a transit to/from certain central African countries. Furthermore when the origin is Africa, there is no requirement to return to the country of origin - anywhere in Africa will do as the end point. So, I suggest you substitute NBO for CPT at the end.
Also note that

• because Europe is no longer the continent of origin, you are not resticted to 2 stopovers there, and so no longer need to transit LHR and MAD
• also there is no restriction on you stopping in Sydney twice if you want
• you have one segment left, perhaps it could be used in SthAm to sort out the getting to/from USH, alternatively you could fly AKL-BNE-SYD instead of AKL-SYD
• it's now a *ONE5 because you've added Africa