Succession in the Ottoman Empire went to the eldest male of the dynasty. Your POD is just after the overthrow of Selim III, so Mahmud II would be Sultan, and he was actually the LAST surviving male of the dynasty until he had sons, two of whom survived. Abdul Mecid, who was succeeded by Abdul Aziz, his younger brother. Both were reformist, although Abdul Aziz developed autocratic tendencies. He was overthrown for overspending the empire into bankruptcy, and replaced by Abdul Mecid's eldest son, Murad V, who was in turn removed due to insanity and replaced by his brother Abdul Hamid II, a very strong and reformist Sultan.
The Ottoman Empire was relentlessly reformist from the 1830s on, contrary to Western contemporary attitudes.
Interestingly, Abdul Aziz'z and te Khedive Ismail's mothers were sisters, so they were cousins, although that has no dynastic implications because the commonality was through females.
Egypt can't really become independent without a chaotic and destructive result, but it can remain autonomous within the empire. Egypt by the mid 1870s controlled the Sudan, parts of Northern Uganda, Eritrea, and the Somali coast to Berbera.
In the Balkans, the Ottomans in your "present" controlled Thrace, Macedonia, and Albania, with Bulgaria autonomous. The portions under Ottoman rule are about 60/40 Muslim, with parts of Epirus Greek majority. There is nowhere other than the province of Janina (Greeks) where Muslims are not the largest group. Macedonia had more Christians than Muslims, but is divided between several different and competing ethnicities. If the Ottoman domains were partitioned among dominant groups or faiths, the Ottomans would have retained all of Thrace and Macedonia up to Salonika (which actually, strangely, had a Jewish majority).
The reason it is important that Libya remain Ottoman if you want independent Muslim states, is that the Ottomans, allied with the Senusi order, were the only forces resisting European colonization, and only the Ottomans would be willing and able to provide military equipment and training to these states, which were all Muslim.
Your best candidates for statehood are Wadai (just to the West of Darfur), and Bornu (on the West of Lake Chad). Zanzibar could work as well, although likely as a British protectorate, unless strongly supported by Egypt and the Ottomans. Zanzibar controlled not only the island itself but the coasts of today's Tanzania, Kenya, and parts of Somalia, and Zanzibari traders and slavers dominated the interior beyond the coastal strip.
Justin Pickard said:
I don't know much about the state of the Ottoman Empire around this time. I'm probably going to have it gradually edged out of Europe by the Greeks and a pan-Slavic movement. Mind you, in the longer term, Russia is going to be as much of a threat. Could we have Abd-ul-Mejid (1823-61) marry, and father an modernising/reformatory heir? And you are right, Libya should be Ottoman.
I really want an independent Egypt, in the longer term - could the Ottoman Empire survive without it? I think Egyptian history will remain more or less as per OTL until the reign of Isma'il Pasha (1863-95) who invests more prudently and sustainably in Egyptian industrialisation, avoiding courting the debt of OTL. I think that a less extravagent personality will also allow him to keep a safe distance from the affairs of the European powers. Ergo, no European intervention, and no dual control.
I'm not so sure about Ethiopia, as I don't know enough about its history...