My understanding is that this was Emperor Haile Selassie's doing to better align the church to the monarchy since his legitimacy was in part due to the church, and his interest in African politics diverged his interest away from Nasser's Egypt which was more focused on the Middle East. Around the time of the autocephaly the Egyptians didn't even bother to have a treaty about the Aswan High Dam with the Ethiopians which should well demonstrate how far apart the two were from each other.

This split in focus between the Middle East and Africa is also related to why Selassie is considered the messiah by the Rastafari and not, say, King Farouk. Although the major consequence of the autocephaly was that now Selassie had financial control over the church and therefore pretty much its full support.

I'm sure that you are right that there were major church politics between the conservative and more liberal factions of the church, but the primary impetus was Selassi's African political interests after his return to the throne after WWII.

For a larger view of the Egypt-Ethiopean political split you should look into Egypt-USSR and US-Ethiopian relationship, The Suez Crisis of 1956, Egypt's and support of the Somalians and Eretrians against the Ethiopians.

Hopefully someone who is more of an expert about the cold war politics of the region can give a less cursory answer for you and can dive into the internal church politics. It's actually pretty surprising that there wasn't a schism, but that would have probably been too radical for the Solomonic dynasty's ancient relationship with the church.