Are we allowed to go before 1900?
Have Thewodros II manage to keep Menelik around, and enforcing the lad's marriage with his daughter and eventually naming Menelik heir.
Have Empress Tewabech surive, or Thewodros take her death much better and not detoriate in mental health.
Have the Foreign Office reply properly to his letter to Queen Victoria and some missionaries and perhaps other people arrive by invitation. Have Thewodros send Menelik and Alitash Tewodros (Thewodros II's daughter) to represent him in Britain - they get some education, marvel and the wonders of modern society and return eager to reform.
The Ehtiopians slowly reform during the reign of Thewodros an then faster during the reign of Menelik II. One of the last actions of Thewodros II before he dies is to declare war on the Mahdi in support of the British. While the Ethiopian army prove woefully incapable in the fighting against the Mahdi's Sudanese troops, the ability to send supplies and some reinforcements of Ethiopian troops down the Blue Nile allows Gordon to hold Khartoum until the relief expedition arrives.
Grateful, the British build a railroad between Massawa and Gondar and Gondar and Khartoum.
The Ehtiopians themselves slowly and painfully build railroads from Gondar to Magdala and from Magdala to Addis Ababa.
Ethiopia gains Eritrea as an autonomous part of its Empire, but with British basing rights. In practice, Ethiopia is a protectorate of Britain and viewed as a decent proxy to control the Horn of Africa. Thewodros II and later Menelik II secures the Ogaden and Djibouti and forces the Sultans of Somalia to pay tribute, although full control is always out of their grasp.
Under British pressure, Menelik II ends slavery 1895, and crushes the massive noble and church revolt 1896 with his modern army equipped with Lee-Enfield rifles, Vickers machine guns and artillery and even an armoured train!
At the funeral of Victoria 1901 and the coronation of Edward VII 1902, Emperor Menelik II attends. The exoticism of the Ethiopian monarch and his retuniy creates great press and the insistance of Edward VII that "either he's a god-damn kaffer and does not belong at all, or he's an Emperor and shall be treated as such" when the Crown Prince of Germany protest at being placed after Menelik in rank (OTL he said the same about the King of Hawaii IIRC) does a lot to endear the British among the Ethiopians.
During ww1, the Ethiopians send a division to aid the British in Egypt and during ww2 they send a corps and have a tank brigade equipped by the British.
Thus after ww2, Ethiopia is a "Federal Empire" including Djibouti, Ethiopia, Eritrea and parts of today's western Somalia, the riches country in Africa after South Africa, a close friend (and perhaps also a member) of NATO, a stable constitutional monarchy and democracy with a decently developed infrastructure of railroads and roads, which allow it to counter starvations and droughts.