They would obviously have to colonise before the Scramble for Africa to avoid conflict with Britain, but doing that would be difficult. My personal suggestiom for a scenario is this:
Since there is a small window of time between the Congress of Berlin and the British Invasion of Egypt, let's assume for some reason the Ottomans are so angry at AH for occupying Bosnia they declare war. They obviously lose after foreign intervention, and AH gains a small area in the Nile Delta. Britain would be worried due to the Suez Canal being there, and it's likely they would both go to war. Let's say AH manages to conquer Egypt before the British have a chance to act, and they make peace. Moving forward, this would probably not affect the Scramble for Africa that much. When WW1 breaks out, though, things would be very different. The bad relations between AH and Britain could lead to an earlier DoW before the invasion of Belgium. The African fronts will also become more important. The Ottomans could focus on only Mesopotamia and the Caucasus, and the Central Powers have the Suez Canal from the start. My guess is that Austria loses badly in Egypt, after being attacked from every side, only holding on to Suez with German and Ottoman reinforcements. In the end, we would probably see the war end slightly later, but with largely the same results.
