Note: This should by no means be taken as a comprehensive timeline. It is, rather, an outline intended to hit the key points of technological development and historical alteration. All named individuals are actual historical figures.
8th-13th C: Golden Age of Islam happens on schedule under the Abbasid Caliphate.
13th-14th C: Mongol Empire smashes the Caliphate government structure, though they don’t come far enough west to really destroy the north African cities and libraries.
15th-16th C: The Ottoman Empire continues scientific development. Selim II’s Grand Vizier, Mehmed Sokollu, succeeds in convincing him to support the Morisco Revolt in Grenada instead of conquering Cyprus, so the fleet is not lost and money is not as severely drained replacing it. (The treaty that ends this supported Revolt renews the grant of religious tolerance in Spain.) Relative continuing prosperity opens the door to some innovations. Taqi al-Din’s engineering work receives government attention in addition to the astronomy and prediction regular for his position as court astronomer. In particular, his steam turbine is applied to propulsion of ships. This re-invigorates interest in scientific development and the value of the the polymath tradition, and leads to resuscitation of many of the Golden Age ideas. Cairo and Constantinople are both centers of study, and the revival spreads through the north African region.
17th-18th C: The Ottoman Empire and north Africa in general exchange ideas and advances with Europe and Asia and the Americas through growing global trade. Armed conflict is widespread throughout Europe, the Mediterranean, Asia and the Americas. Ottoman military reform happens on schedule, the suzerain states move toward independence, and the Ottoman Empire focuses on improving productivity and avoiding invasion. There are stirrings of an Abbasid revival in Egypt, as the revival of Golden Age scholarship and texts involves study of pre-Ottoman Caliphate law and politics as well as science.
19th C: Tanzimat (the reform movement) happens on schedule. Thanks to the revivalist trend, rather than Western nationalism per se, a sense of solidarity (asabiyyah) gains momentum in the old Caliphate areas. Egypt successfully resists Britain’s occupation, not least because of the products of the engineering centers there. The Ottoman Empire, through Egypt, starts spreading transport and weapons through northern Africa to create alliances there that will secure that flank against further attempts. Once the trade is going, sub-Saharan nations like the Oyo Empire and the Igbo nation and the Zulu kingdom all want in on it, and Africa is armed in time to make the Scramble for Africa considerably less successful. In the process, it is possible that Egypt, Lybia and Algeria successfully break away from the Ottoman Empire and serve as the nucleus of a new Caliphate and new technological center.