I don't think it is likely that Britain would ever manage to control the entire continent; other powers had trading stations and ports on the Continent dating from the Golden Age of Discovery, and the French got into Algeria relatively early, and would be hard to displace for a variety of reasons. However, if one excludes the pre-existing European enclaves and French Algeria, I do think it would be possible for Britain to gain the majority (60-75%) of Africa in the period between 1815 and the 1870's when Britain's European rivals would finally have the money and motivation to stake claims to new land in Africa.
Expanding north from their base in South Africa, the British would not have run into much into much hard resistance aside from the Zulu. Although dry, Southern Africa (OTL South Africa, Namibia, Zimbabwe) also mostly lacks the deadly tropical diseases that kept out Europeans. If the English, for whatever reason were to battle the Zulu in Post-Waterloo period, and defeat them, they would have a clear path to dominating Southern Africa. A political accommodation with the Boers would make this much easier. The Zulu were formidable in battle, but their weaponry was too primitive to stand against disciplined musket fire, so as long as the British were able to stay in tight columns and quickly form up into infantry squares, they could probably avoid the disasters that accompanied OTL's Zulu War.
The next logical area for expansion would be East Africa. That part of the continent was largely dominated by petty kingdoms and Arab sultanates, the most formidable of which were Zanzibar, and the Omani Arabs. The British had the naval capacity to clear the Indian Ocean of their navies, it would be within the capacity of the RN and Royal Marines to force their suzerainty on the local elites through a combination of force and other inducements.
Beyond the Coast of East Africa, the next major obstacile would be the Abyssinian Empire. Before than country had the opportunity to import European weaponry they would have been an easy target. The British had enough troops in India to take them on during the middle of the 19th century, the problem would be how to borrow them from the EITC, and equipping them. In fact any British conquest of East Africa would probably rely heavily upon Indian raised troops such as Gurkhas and Sikhs, which could have interesting butterflies.
With the control of East Africa to the borders of Egyptian Sudan, I think the British would be inclined to stand pat until at least the 1870's. During that time they could gain de facto control of the Khedivite of Egypt, and thus control the entire East coast of the continent, from Alexandria to Cape Town. Barring the discovery of gold, there was little to attract Europeans to the interior of the continent, and the tropical diseases were far too deadly. In fact, I can think of no rational motive that would induce the British to expand into Central Africa (the Congo) or Northwest Africa other than national pride, or fear, at least until medicine had advanced a bit.
However, explorers would eventually penetrate the interior of Africa, and discover sources of wealth to be exploited. Starting in the Great Lakes Region, the British could expand into the heart of the continent, until they reached the Kingdom of the Kongo, another hard nut to crack. Both the logistics of operating that far from the Indian Ocean, and supplies, as well as the presence of deadly diseases which would kill both man and horse would make a sustained campaign difficult. Reliance on Indian troops and local auxiliaries could reduce these difficulties somewhat.
I leave it to others to contemplate how the British could extend their conquest of (most of) Africa from South of the Congo River to encompass what became French West Africa. I cannot even think of any reasons the British would have to tangle with the Taureg nomads of the Sahara, other than abolishing slavery (something even the present governments of that region have not succeeded in doing)