Due to the length of this post, I'll break it up by city pairs – Johannesburg, South Africa and Dakar, Senegal then Capetown, South Africa and Accra, Ghana. Though there are lie-flats available to Lagos, Nigeria (LOS), I will be omitting that section since the US Department of State considers criminal threats in the area "critical".
Methodology for this article: I first checked Flight Stats for Departures and Arrivals at these particular airports to gauge what my options were. Then I matched the results with Seat Guru to see if I'd get a lie-flat option. From there, I had to figure out how to get there using various awards. Here is what I found.
To Cape Town, South Africa (CPT)
Fly Lufthansa from North American city to Munich (MUC) then continue MUC to CPT.
This route is seasonal though the A340 offers lie-flats in old First. You can book with 150,000 United MileagePlus miles if there's availability and can book within 2 weeks of departure.
Fly British Airways from North American city to LHR then continue LHR to CPT.
First and Club World Class both offer a lie-flat seat. Very sparse availability found for both First (100,000 AA miles OW) and Business (75,000 AA miles OW). Also note that American Airlines assesses the high fuel surcharges on British Airways.
Fly Virgin Atlantic from LAX to LHR then continue LHR to CPT.
Upper Class has fully flat beds. It costs 220,000 (100,000 miles LAX-LHR; 120,000 miles LHR-CPT) Virgin Atlantic miles with huge surcharges.
To Accra, Ghana
Fly Emirates from North American city to Dubai (DXB) then continue DXB to ACC.
Both First and Business offer lie-flat seats. According to the Emirates call-center because this route is flown on a "not so advance" 777, First does not offer private suites.
You can book this with Alaska Mileage Plan via Alaska Airlines Reservations at 1-800-252-7522 (TTY/TDD: 1-800-682-2221) , daily between 5:00 a.m. and midnight (PT). I called to check availability for February and March but the call-center said it wasn't open yet despite usually having availability 330 days out. I asked her to check earlier availability for October or November but no First/Biz availability. It costs 200,000 round-trip for First and 155,000 miles round-trip for Business.
You can also book with JAL. Lucky details how to book Emirates with JAL and Starwood here. Since they use a distance-based award chart, I looked up routing from LAX-DXB-ACC which yielded 12,249 miles one-way on Great Circle Mapper. This falls into the 20,001 – 25,000 mile range where First Class costs 200,000 miles round-trip and Business costs 125,000 miles round-trip in Business. I called JAL and they confirmed 200,000/125,000 in First/Business for the 24,476 miles traveled round-trip.
Fly British Airways from LHR to ACC.
First (100,000 AA miles one-way) and Club World Class (75,000 AA miles one-way) both have lie-flat seats.
Fly Virgin Atlantic from North American city to LHR then continue LHR to ACC.
Upper Class has fully flat beds. It costs 180,000 (100,000 miles LAX-LHR; 80,000 miles LHR-ACC) Virgin Atlantic miles.
Conclusion: British Airways and Virgin Atlantic have ugly surcharges all around. Capetown is easily attainable if you're ok with that. Emirates availability was pretty crappy when I called and Lucky recently confirmed this.
Have you redeemed miles for a lie-flat to Africa? Feel free to add onto anything else I missed.