First, let's consider the distance. It's about 4,500 Kilometers/2,790 Miles from Cairo to Pacuba, Uganda on the shore of Lake Albert, the source of the White Nile. That's about the same distance as between Los Angles and New York City. But travelling through the region was much more difficult than across the US: most of the territory was not friendly, if not downright hostile. As it was impossible to travel other than on foot, supplies had to be somehow bought en route and bearers hired in that hostile territory, as it was impossible to carry enough. Add tropical diseases, difficult passages through rough country, and even shorter trips on the continent were very hard. When Speke and Grant struck out from Zanzibar to Lake Victoria, it took them from October 1860 to July 1862 to cover about 700 miles.

Now let's consider the physical obstacles on following the river. If you wanted to go by boat, there would first be the Six Cataracts, between Cairo and Khartoum. These pretty effectively blocked free-floating boats. When the British went to relieve Gordon in Khartoum, it took them months to put together an expedition up the river, including having special durable transport boats made. Admiral Charles Beresford had to rig lines and winches to pull them past the rapids and over rocks. Going from just Wadi Halifa, right on the Egypt/Sudan border, up to Khartoum took three months.

Last, below Khartoum there was the Sudd, an enormous marsh extending over 57,000 sq. kilometers /22,000 sq. miles. It was a constantly shifting landscape and waterscape where it was impossible to tell a channel from a dead end. Even the Romans knew to give up at the Sudd.

With all this, you can understand why explorers like Speke, Burton and Stanley didn't try to follow the Nile down. They struck overland from the east. Even then, they had a very hard time of it.

A very readable ( if somewhat out of date) account of the Nile expeditions is Alan Moorehead's The Blue Nile and The White Nile.