From 1882 until 1914, Egypt was nominally part of the Ottoman Empire, but this was really in name only, as the British exercised de facto control. I'm less interested though in the actual way things were run by the British, and more interested in what the technical structure was that everyone I guess pretended was still the case but obviously wasn't! How was this all organized, officially?
Officially, Egypt was still an Ottoman province, as you mentioned - there was even a token tribute sent off to Istanbul every year.
The government system remained largely intact -- i.e., there was a Khedive, a council of ministers headed by a prime minister, government ministries and agencies and the like, however the power behind the throne was the British consul-General (Lord Cromer between 1883 and 1905), who called the shots, and basically ran affairs out of the British High Commission.
The British and French had taken over various government ministries in 1876 when Egypt first defaulted on its loans; these included the Ministry of the Interior (which oversaw agricultural production; the Europeans being eager to make certain that agricultural products that would fetch high prices on the world market were being grown efficiently and exported in the highest possible amounts) and the Ministry of Finance. It was, in fact, growing internal opposition to the European power over the Egyptian government that led to the military revolt under Col. Ahmad Urabi in the winter of 1881-82 that resulted in the British military incursion and occupation (whoopsie).
During the occupation, all serious affairs outside of the Ministry of Waqf (religious endowments) and similar internal organs were run by the British--Europeans were either the ministers themselves or "trustworthy" Egyptians (usually western educated) would be installed, almost always with a European advisor or vice minister to make sure things were done properly. Servicing the debt was the top priority, and government services were reduced in order to make Egypt profitable (expenditures on education and public health, for example, did not exceed 1% of the national budget until after Cromer's tenure ended).
They de-emphasized industrial production, encouraging Egyptian companies instead to produce raw materials that could be sold to European factories (and then re-imported to Egypt as finished goods); they pushed agriculture in the direction of producing cash crops like cotton--even though Egypt produced high quality wheat and rice, these would then be exported and cheaper commodities would be brought in from India to feed the local population; the Suez Canal was operated wholly by the British and revenue from it went to its European operators.
The Khedive had no real choice but to go along with it. Ahmad II Hilmi got a bit feisty and hoped that WWI might lead to the Ottomans removing the British; he was on a visit to Constantinople when he was deposed by the British and replaced with his nephew Hussein Kamil - shortly thereafter Egypt was officially made a protectorate and Hussein Kamil became "sultan."
So, on the surface, it wasn't the governmental structure that changed so much as who was actually installed into key positions within those structures, how the government was run, and what their priorities were.