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(In case the abbreviation from the title didn't tell you, this takes place in EdT's Fight and Be Right TL)
How could the Abyssinians have taken Khartoum from the British? What effect would it have had on the rest of the War of the Dual Alliance if they had taken it?
I can see it happening a couple different ways:
The British and the Abyssinians aren't as cordial to each other as they were OTL, the officers don't visit each other and so the British aren't able to get the exact location of all the tents and stuff, which means that the artillery strike doesn't completely break them.
Instead of waiting around, the Abyssinians attack Khartoum as soon as they hear that war was declared. I'm not sure if they would have been able to take control of the city by force, though.
Maybe there's less food in the city and they're able to successfully starve them out?