In general, the breakdown of the Ottoman order by the end of the 18th century, resulting in the problems with local elites (ayans) followed by ethnic disorders in Serbia and later Greece, was in large part the rest of stresses caused by the wars with Russia from 1768 onward. With the more peaceful and prosperous Russian-Turkish relations of TTL there's no particular probability of the events resulting in the Serbian and Greek rebellions and the disorders that followed such as the breakaway of Egypt.
Actually while Russia did play the part in breakdown of the Ottoman order it was not the only actor and internal problems of the Ottoman empire can't really be ignored (otl Austrian backed renegade Janissary Osman Pazvantoğlu was defeated by Serbians under Sultan's banner so there were internal problems with Janissaries and with, or without Russia empire was in need of the reforms and Janissaries and Serbians were bound to clash which was a cause for Serbian revolt otl) . Regarding Serbia, without Russia Austro - Turkish war of 1735-39 never happened which means that Serbia remained under Austria for much longer (over 50 year's) and was enjoying relative autonomy and economic prosperity (same thing happened otl in much shorter time and then Serbian population jumped from around 120ks to 400k and Belgrade became Ottoman second largest city with population over 100k, over fifty years of Habsburg rule left even bigger impact ), so when Ottoman empire attacked the Austrians in 1771-72 and regained those lands it was quite a shock for the Serbs to return under the Ottoman rule and given that Serbs were always cooperating with Austrian invasions Ottomans couldn't simply return things to how they were thus there was need for autonomy (at the time of war of 1771-72 Mustafa the III was the Sultan and that guy did some military reforms otl, he was Fredericks fan and started a war against Russia otl, in TTL relations with Russia were good so Austria exhausted from seven year war (given that Russia stayed out of it) was obvious target ). Regarding Egypt? Sultan's peace there was actually disturbed by the Napoleons invasion, something that happened otl and had nothing to do with Russia.
As for the Greeks? While Greek revolution was largely inspired by the Serbians it is important to note that they were inspired even more by the French revolution and had their base in Greek diaspora despite Greeks having relatively good position in the empire (entire upper echelon of the Church was Greek, they were part of Ottoman administration and were controlling good part of the trade) .