Eek, this is way after the event. But alas there is no commonly accepted size tyre we found when looking at ALL the countries on the east coast.
In summary our experiences were :
Egypt - there is 1 (one) Michelin dealer where you can buy 285/75/16. It's in Hurghada! You can find anything, quality varying, in Cairo
Sudan and Ethiopia - the standard tyre is NOT BFG / anything fancy / specialist tyres. No AT's or Muds. All tyres are pumped HARD road tires! Standard size 265/x/16
Used to gun it at hair raising speeds over bumps and all kinds of crazy terrain. They simply can't afford special tires and they use a set every return journey on the "Moyale road" so no use in buying special tires so the locals said.
Kenya - you can get everything in Nairobi. But that's a city in a large country. Outside of that, it's the standard MO ie. road tires pumped quite hard. maybe AT's (usually worn / second hand)
Uganda - We actually bought a tyre here. The biggest they could find in what we think is the most advanced country apart from SA in Africa was a 265/70/16
Not even close to ours! But it was rubber attached to a round metal spinning disc so was perfect
Rwanda - Make sure you have your own spares. Everything is shipped in by road from Tanzania and is very expensive. £200+ a tire at least for a new one. You'll get whatever they can find..
Tanzaina - Everything available in Dar Es Salaam. Again, it's a city in a massive country. Same as Kenya.
Malawi - Have your own. It's close enough for them to get you something from Lusaka, Zambia or Tete, Mozambique but you'll pay. Not many cars on the road. Not even many donkey carts here. National speed limit of 80kmph
Mozambique - Pretty sure you could get what you needed in Tete or Maputo. Tete is in the sticks so you'd wait, and with Africa - pay for the priviledge.
Maputo. Hell on earth. Lots like Dar! but you can get most things here nowadays and nothing is more than a few days from Johannesburg so you're on the home stretch. It's a 4 hour odd drive to the SA border [safety

]
SA. Everything and everything like the UK. Again. Be prepared to pay. Last I heard in June 2011, 1 x BFG KM II cost £300 a tyre for 285/75/16
In summary:
We took 5 tyres in total, ie 1 spare.
They were BG KM II's 285/75/16
We lost the spare / it fell off / was stolen
We had just one puncture in Uganda in the city. I had a repair kit but at $1 to fix it by an expert who fixes tham all day it was worth it. Took him 30 seconds!
Whilst we did not carry the African "standard", tires have come a long way and the roads are generally good. If you plan on doing more extreme stuff, I'd question whether you're still overlanding

Given the advantages of the 285's over 265's in the long run, I'd still suggest them.
- better longevity post trip, bigger tires generally sail over obstacles smaller ones wouldn't etc.
I would strongly challenge anyone who thinks 2 or more spares are needed. There are many other limitations on your truck and whilst tyres are critical. There are many other things that can render you immobile. More is not always better
Definitely take a repair kit. Invaluable at £15, light weight and small in size.
Got a Silverline one off eBay. Make sure you get the offroad one not the car one.
Everyone we encountered who was a local ran road tyres pumped hard
Used re-treads, and mostly parts from China
They live there and manage just fine!
Standard size was 265/16/75
I love my KM II's.. I swear by them and would take them again in a heart beat.
Good mileage, no noise, good grip and they look smashing
(The only other I'd suggest are BG At's. They were the most common "fancy tire" we encountered. Better mileage, but they don't survive as well on the axle shattering Moyale road. A strong contender..)
Bit late but perhaps will help in the future..