It seems that you are positing that the cost of methane, fresh water, and electricity will plummet in the region around the Sahara. Based on this premise alone, fossil fuels would not necessarily disappear. The most expensive fossil fuels (e.g. oil shales) would be shut down, but existing, land-based oil wells are extremely inexpensive to run and would continue to produce oil even at very low market prices.
An inexpensive source of alternative energy would make it much easier to pass anti-fossil fuel legislation in an effort to reduce climate change. I will assume that such a law goes into effect, and that it is enforced successfully across the entire globe.
When wealth comes from fixed resources which are tied to geography, the result is usually an oppressive regime and tremendous wealth/power disparities.
In the feudal era lords became the government because they controlled agricultural land. They did not control the tradesmen in the cities because those people were highly mobile and could take their wealth generating ability with them.
In the modern era most nations where mineral wealth represents a large proportion of the GDP come to be governed by whatever small group of people control that mineral resource (often oil). This is the case in Venezuela, Saudi Arabia, Russia, The Central African Republic, etc.
If a fairly small group of people is in control of this Sahara Project, it will rule in an authoritarian manner. Employees and other residents of the area will not have strong rights such as freedom of speech. Owners will appoint the judges or whatever arbiters are used for the legal system. There will be no rule of law governing disputes between owners and everyone else, as the owners will always win every dispute. There might be rule of law governing disputes between non-owners though.
The ownership class will get away with rape, murder, assault, and theft. Their children will be enormously wealthy and immune to law, but will often have no real power until their parents reach retirement age. Held in a state of perpetual adolescence their behavior will be terrible.
Politics among the ownership class will be cutthroat as everyone will jostle for power in the absence of an objective rule of law.
Despite what you say about there not being locals to displace, every piece of land (with the possible exception of Antarctica) has people who consider it theirs. The owners might be miles away, but they exist. It is very likely that the project would proceed as follows:
A large foreign company (backed by massive injections of private capital) approaches the local people of each region of the Sahara and negotiates a deal with them. Normally the locals would receive little compensation in these types of contracts, but foreign lawyers rush in to represent the locals for no cash up front but a huge share of revenue. Thus the locals are able to negotiate very lucrative joint ownership contracts with the company.
In the ensuing years the cleverest locals are able to buy out other locals using a wide variety of clever business tactics. When ownership finally stabilizes, a very few families control most of the land and become the de facto royalty (similar to the establishment of the house of Saud). The rest of the local families form a long tail of mid level and small time nobles - much wealthier and more powerful than workers, but trivial compared to the royalty.
The ownership class will consist of the native land holders, major company shareholders, the company board and officers, and the lawyers who helped the land holders negotiate the deal.
Labor, both skilled (there will need to be engineers to install, maintain, and troubleshoot the facilities) and unskilled, will be non-citizen resident workers. They will have few rights, and will not be allowed to retire in the Sahara. Their children will not have citizenship. Any adult without a job (or a spouse with a job) will be forced to leave. Any child without employed parents will be forced to leave.
The Sahara company will need to run its own military as the facilities will be very vulnerable to sabotage, terrorism, and military attack. Pipelines and power lines can be bombed. Solar panels can be destroyed by nearly anything. Electrical substations in particular are extremely vulnerable to attack.
If the Sahara can do this, then other deserts would likely be almost as good and would replicate this economy. Expect large rival projects in the Atacama, Australia, southern Africa, the Namib, Mexico, and the US southwest. The Gobi will be able to produce huge amounts of power, even if water is not convenient.
The old oil powers will experience upheaval. Smart families will be among the investors in the Sahara company (and its competitors). Stupid families and nations will just become poor and irrelevant. Oil will still be used for plastics and lubricants, so it will not go away, but its price will plummet. Plastics, in particular, will become extremely inexpensive.
The governments of the oil nations will be overturned. These nations will become much more normal, with governments which are controlled by a wide variety of industrial and commercial powers rather than just a tiny minority of oil powers. The behavior of these nations would become less outrageous as they adopt policies more in line with what makes business leaders happy. (i.e. stability, rule of law, free markets)
Coal has never comprised a large fraction of any nation's GDP and its collapse will not result in major changes to any nation state. Coal will have essentially no use in a post-fossil fuel economy. The same goes for natural gas.
Norway and Alaska will face financial crises and be forced to cease their negative taxation programs.
You mention the electricity being sold to Africa, but a large portion would also go north, under the Mediterranean to Europe. The cost of power increases with the length of transmission lines, so there would be a gradient of electricity prices based on distance from deserts.
Wherever electricity is cheapest, massive aluminum smelting operations will be established. If the average price of power across the globe decreases, then so will the cost of aluminum (as well as titanium). Aluminum smelting operations produce enormous quantities of waste heat, which could presumably be used in the already existing water purification plants.
Having massive quantities of cheap, fresh water available in areas with plentiful sunshine and warm temperatures would result in an ideal agricultural environment. The price of food should plummet as Saharan vegetables become common ingredients at the dinner table.
The change in albedo and evaporation rates this would produce in the deserts would cause unpredictable climate changes. Winds would move a little differently, and rainfall would happen in different places. In general though, the net result would be a slightly wetter, not a drier world - especially in the areas around the desert zones.
It is very unlikely that the national borders of the desert regions would be redrawn, so I think you should assume that the nations containing large amounts of desert would become much more powerful. This would include Egypt, Libya, Tunisia, Morocco, Oman, Yemen, Syria, and Iraq, but also Namibia, and Chile.
I think you should have Saudi Arabia be one of the smart oil powers which smoothly parlays its oil wealth into Sahara Project investments.
Russia seems likely to wind up a very big loser in this transition. It has little access to solar power, and has hobbled its industrial and commercial economies with crazy oil-state authoritarianism. This could be awkward as they still have one of the largest nuclear arsenals in the world. Perhaps the US and China restart the Clinton-era policy of purchasing Russian fissile material from old nuclear warheads? (If you're not aware, during the 1990s the USA purchased huge quantities of uranium and plutonium from old Soviet warheads, and then burned this material in US power reactors. In exchange Russia gained much needed cash during their transition from communism.)