For a few reasons.

First, while the video mentions the 1889 Urabi uprising, it forgets to mention the 1919 and 1921 Cairo uprisings.

From 1889 to 1914, Egypt was still technically considered a part of the Ottoman empire (though an area that had been at odds with Istanbul since 1805). This period was known as the "hidden protectorate" where Britain tried to keep a low profile to preserve the diplomatic fiction (a) that Egypt was not irrevocably taken away from Ottoman empire and (b) that the Pashas of Egypt were in control of their own country.

But, with the Ottoman Empire joining the Central Powers during World War 1, Britain was forced to change policy. Britain recognized the Egypt as legally separate from the Ottoman empire in 1914, and formally engaged in a protectorate agreement with Egypt. 1914-1922 is known as the period of the "open protectorate", because the relationship was publicly acknowledged.

During World War 1, this situation was tolerated (and dissent was muted by a wartime martial law decree), but with the diminishing of the Ottoman threat and return of peace, Egyptian nationalist irritation with British military presence (occupation?) in Egypt grew, leading to anti-British demonstrations in 1919 and 1921. These protests were suppressed using armed force, leaving protesters dead.

These protests and the response generated enough ill-will among the Egyptian public that the British government appointed a commission of study, which recommended formal Egyptian independence and treaty agreements that would allow continued British presence. Britain did issue a Unilateral Declaration of Independence in 1922, but the UDI included a reserve clause that ensured the status quo for legal extraterritoriality of foreign citizens, British military deployments, control over the Suez canal, and the status of the Anglo-Egyptian Sudan.

It was intended that these points would be addressed with the quick negotiation of a treaty, but no such treaty was agreed to until 1936. Because of the threat of the 1935 Italian invasion of Ethiopia, the Egyptian king Fuad agreed in 1936 to Britain withdrawing soldiers from Egyptian cities, and the presence of 10,000 British soldiers in the Suez canal zone until 1956, when it would be determined if Egyptian army could provide security for the canal and replace British presence. In the event of war with Italy or another foreign power, Britain could increase their troop presence for the protection of Egyptian sovereignty.

So, when World War 2 happened, Britain did massively increase the presence of British army, navy and air force assets in the canal zone and throughout Egypt, to protect against German and Italian military threat. However, while Egypt did formally declare war on Germany, the Egyptian government and military stayed largely neutral in the conflict (while allowing/facilitating British logistics), and military and public opinion treated British presence as yet another occupation.

So, all of this is to say that the British would have known that any effort to deploy British troops in Egyptian cities in 1952 would have provoked further anti-British sentiment, and weakened the Egyptian monarchy by making them look like a British puppet.

---

Further, we shouldn't get caught up in hindsight-bias. As the coup was happening in August 1952, the British might have had an idea from the public pronouncement of the coup leaders that the officers presented themselves as patriotic and opposed to the corruption of King Farouks regime. The British probably did not have a detailed idea of what the personal politics of Ngoubi, Nasser, Sadat or the other officers were, or what that implied for policy items like the Suez canal or the Sudan.

And, given that the Wafd party had won the 1950 parliamentary elections and in 1951 had declared the abrogation of the 1936 treaty (i.e. insisting the Brits get out of the canal zone), a change in government may have looked like an opportunity to try and re-negotiate the terms with a new government.

In fact, Britain did negotiate an agreement with Gamal Nasser in July of 1954 that called for the phased withdrawal of British troops over two years until June of 1956, but allowed continued presence of British civilian maintenance staff and the possibility of re-deployment of British troops in the canal in the event of war with the Soviet Union.