From September 1st 1939 to September 2nd 1945 (the conventional dates for World War 2), Egypt could best be described as a condominium. The definition of a condominium is below cited from this article:
Regarding international law, "condominium" refers to territory that is governed by multiple sovereign powers who have formally agreed to share duties without necessarily dividing the area into national zones.
The British controlled communications, foreign affairs, the suez canal, and legal jurisdiction over their own soldiers in camps while the Egyptians controlled everything else. Hence two governing bodies were sharing duties over the sovereignty of Egypt.
What Egypt controlled and it's status of a state from September 1st 1939 to September 2nd 1945 meant different things and abruptly changed. It depends on what part of the war you talk about as well as which point of view you consider.
To begin, the Unilateral Declaration signed in 1922 that you cite stated that Egypt did not have control of foreign relations, communications, and the military as stated below:
- The following matters are absolutely reserved to the discretion of His Majesty's Government until such time as it may be possible by free discussion and friendly accommodation on both sides to conclude agreements in regard thereto between His Majesty's Government and the Government of Egypt:
(a) The security of the communications of the British Empire in Egypt;
(b) The defence of Egypt against all foreign aggression or interference, direct or indirect;
(c) The protection of foreign interests in Egypt and the protection of minorities;
(d) The Soudan.
Britain thought of it as an independent country, but Egypt did not have full control. From the perspective of International Law, this fits the definition of condominium. This changed slightly during the Anglo-Egyptian Treaty of 1936, where the British withdrew troops except those at the Suez canal and agreed to train Egyptian troops as stated below on Page 19 of the treaty:
His Majesty's Government in the United Kingdom will furnish the Military Mission which the Egyptian Government desire, and will also undertake to receive and provide proper training in the United Kingdom for any personnel of the Egyptian forces which the Egyptian Government may desire to send for the purposed of being trained.
The essential status of condominium was not changed (from the International law point of view); however, Egypt retained more control over it's land, save for the Suez. The British in return spent the effort and provided resources for training Egypt's troops. This essentially lasted through both the Italian and German, invasions where, for a brief time, parts of Egypt could have been considered an occupied state. But the British victory at El Alamein handily prevented that. I do not think that the Abdeen Palace Incident of February 4th 1942 changed its political status either.
From February 4th 1942 all the way until the end of World War 2 (September 2nd 1945), no events occurred that fundamentally changed the political status of Egypt from a condominium.
 I am leaving out anything related to Anglo-Egyptian Sudan since I assume this question deals with the polity that is the direct ancestor of the country we know today as Egypt.