It’s all largely a matter of personal preference. Let’s rule out lever actions, because they usually aren’t chambered in calibers suitable enough for African dangerous game (barring the .405 Winchester and certain wild cat cartridges). On the subject of the .405 Winchester, the factory loaded ammunition which I’ve seen for this caliber (Hornady) doesn‘t offer good performance on Nilgai sized game. On the other hand, hand loads (with Northfork bullets) turn it into a very competent dangerous game caliber.
If you have to choose between a bolt action or a double rifle, you need to first ask yourself as to what’s more important: Two almost simultaneous shots or three to six shots (depending on rifle and/or caliber) with a split second interval between each (for bolt cycling) ?
Double rifles have a limitation: They don’t deliver accurate groups with ammunition except which they’ve originally been regulated for. For instance, I’ve seen an Auguste Francotte sidelock ejector in .458 Winchester Magnum which was built in 1969 and regulated for Winchester 500/510Gr ammunition (advertised velocity being 2130fps). After 1973, Winchester lowered their advertised velocity for their .458 Magnum ammunition to 2040fps. In 1980, I had a chance to fire that Francotte double rifle with (then) current production Winchester factory loaded ammunition. Groups were as wide as six inches at 50 yards. So if you opt for a double rifle, make sure that you:
a) Have access to a good stock of ammunition which your double rifle has been regulated for.
or
b) Know how to hand load and replicate the ballistics of the ammunition which your double rifle was originally regulated for.
Double rifles definitely have a very strong place for hunting in terrain, where there is thick cover (such as in mangrove forests or the former Belgian Congo or during follow ups of wounded game). Dangerous game is found at close ranges and the split second it takes to pull the second trigger can decide whether you’re the hunter or the hunted, On the other hand, bolt action rifles are far more suitable for hunting in open terrain where long range shots are offered.
My take is as follows:
For a new client hunter visiting Africa for the first time, a bolt action is hands down the way to go. It’s easier for a beginner to make a more accurate first shot with a bolt action rifle, than with a double rifle (generally speaking). And a client hunter’s biggest priority should always be to make sure that the first shot is properly placed.
For a professional hunter, it’s largely a matter of personal preference. For example, Doctor Kevin Robertson prefers a Dumoulin .505 Gibbs built on a Brevex Magnum Mauser action. Mark Sullivan prefers a Heym Jumbo in .577 Nitro Express. Jeff Rann opts for the pragmatic solution. He prefers a William Evans boxlock ejector .500 Nitro Express for thick cover and a John Bolliger .505 Gibbs built on a Granite Mountain Arms African Magnum Mauser action for open terrain.
These are some of my dangerous game, hunted over the years in Africa & Asia.
.375 Holland & Holland Magnum
.600 Nitro Express
.404 Jeffery
.375 Holland & Holland Magnum
.404 Jeffery
.375 Holland & Holland Magnum
.375 Holland & Holland Magnum
.375 Holland & Holland Magnum
.375 Holland & Holland Magnum
.375 Holland & Holland Magnum
.375 Holland & Holland Magnum
7x57mm Mauser
As can be seen, most were taken with bolt action rifles.