At the risk of appearing pedantic 1882-89 is two wars against quite different opponents for quite different reasons; it's not really legit history to bracket these dates as a single operation of war. Arguably it's actually four distinct campaigns; Egyptian Intervention, Gordon Relief Expedition, Red Sea Campaign, and Defence of Upper (or is it Lower - I never can remember!) Egypt.
And I'm sorry to break it to you Mark but neither the South Staffs nor the Black Watch even got close to Khartoum. Check the map for Kirbekan's location. You may be surprised how far north it is: there is a fearful long bend in the Nile beyond Korti - in a literal sense your heroes were, I'm afraid, rowing away from Khartoum, not towards it. The River Column was a non-event in terms of relieving Gordon, except in so far as it provided the logisitc stockpiles which enabled the launching of the Desert Column from Korti. The only British soldiers who got to Khartoum before Kitchener's campaign was launched were members of the Royal Sussex, and, rather sensibly, they didn''t get off the boat!