Well, if an Egyptian Army field ambulance goes to Korea, presumably they could be attached to the Turkish Brigade (itself attached, generally, to the US 25th Division) in a spirit of pan-Islamic amity, while the IDF battalion is attached to one of the other US divisions (1st Cavalry, 2nd, 3rd, 7th, and 24th infantry divisions, or the 1st Marine Division).
And then how do the Egyptian authorities explain (to their (agitated and voting) population) what happens if an Israeli solider finds himself on stretcher in front of an Egyptian doctor?
More importantly, given that even the Wafd Party was losing popular support because it wasn't nationalistic enough towards the British in the 1930s (a factor I'm sure in their abrogation of the 1936 Treaty in 1951 - a move hinted at from November 1950 and likely aimed at winning back the nationalist base they were losing (and which was temporarily successful, but they unleashed more than they could handle and ended up directing nationalist focus to the total removal of the British from the Suez Canal Zone)) and
given this:
The Egyptian monarchy was seen as both corrupt and pro-British, with its lavish lifestyle that seemed provocative to the free officers movement who lived in poverty. Its policies completed the image of the Egyptian government being a puppet-figure in the hands of the British government. Promoting the feeling of corruption of several Egyptian institutions such as the police, the palace and even the political parties by the free officers. Despite allegations of anti-British sentiment, a CIA document dated 23 July 1952 stated that the dissatisfaction within the army over corruption in high command began in 1948 after the discovery of an arms scandal during the Palestine conflict.
[2] The loss of the 1948 war with Israel led to the free officers' blame of the King and their promotion of that feeling among the Egyptian people.
[2] Tensions between the military and the monarchy resulted in the removal and arrest of Commander in Chief of the Armed Forces Haidar Pasha, Chief of Staff Harid Pasha and other high-ranking officers.
[2] However, scandal subsided over time and the King was able to eventually reappoint Haidar and Harid to their old positions.
[2]
It's difficult to see
who in the Egyptian government would think of sending
any Egyptian military personnel (whether a doctor, nurse or even just a few orderlies) to Korea in the knowledge that an
IDF battalion is already there and a large
British force is there too. That move alone would likely be well used as electoral fodder by the time the 1955 elections were due.
One needs to recall too that Free Officers Movement was established in 1949.
There's
also this (Egypt from Independence to Revolution, 1919-1952 by Selma Botman) (which at this point is discussing Egyptian issues with the continuing British military presence (which also resulted in a violent incident on February 21, 1946 when British troops fired on demonstrators after the demonstrators attacked British armoured cars that were passing through the area of the demonstrations (which until then had been peaceful)) and the Egyptian desire for a British withdrawal from Sudan):
.....Al-Nuqrashi returned to the prime minstry. He took Egypt's case before the Security Council of United Nations in 1947 to appease nationalist feelings in the country and defend Egypt's interests. His voice was barely heard, and the coucil recommended a resumption of negotiations, to the extreme disappoint of all political groups in Eygpt. This type of ineffectual performance made it increasingly clear that the traditional parties were incapable of solving the national question.
Nationalist pride was further bruised by Egypt's performance in the Palestine War of 1948. The population became even more alienated from Egypt's political leadership as a result of the "munitions scandal" in which defective military equipment left over from World War II had been purchased by the palace, high military officials and some politicians and supplied to Egyptians troops fighting in Palestine. The unscrupulous behaviour of supposedly respectable government officials and the embarassing weakness of the Egyptian army precipitated the formation of the Free Officers movement by young, low-ranking military officers who felt themselves betrayed by their superior officers and political leaders...
Egyptian attempts to pressure/battle the British to give up the Suez Canal continued right through 1951 and 1952 and peaked in January 1952. On January 19, Egyptian commandos struck the Tel al-Kabir (the largest British garrison) and a 12 hour battle ensued leaving 50-60 Egyptian commandos dead. On January 25, the British attacked a police station in Ismailiyya (a centre of anti-British activity) and the result was 50 dead Egyptians.
Given the domestic situation you are essentially suggesting that:
1. A King who has
just emerged from the scandal of the failure of Egyptian forces in Palestine and who is regularly seen as being pro-British and possibly a British stooge would be okay with an Egyptian field ambulance being sent to a conflict on the same side as the British and Israeli soldiers
2. That a nationalist party that had just been returned to government in the 1950 elections (before Korea) on the back of nationalist resentment over the loss in Palestine, the continued stonewalling of the British in Suez and Sudan and which was agitating against the British in Suez would agree to send an Egyptian field ambulance to a conflict on the same side as the British and Israeli soldiers.
3. That following on the Munitions Scandal and loss in Palestine that precipitated the formation of the movement that wished to overthrow the King and root out corruption, that the Egyptian government would be able to send a field ambulance to Korea in 1950.
The outcome of such a move is likely outright Revolution and the withdrawal of the field ambulance. So the Free Officers Movement get their ranks swollen by the move, and probably get a figure like Muhammad Naguib to help with a coup in 1950 rather than 1952.
On the other hand the domestic situation and problems with both the British and the Israelis likely means that no Egyptian government in 1950 would be sending any Egyptian forces (even janitors) to Korea in 1950.
As far as how an IDF battalion would have performed, it would be a scratch but presumably volunteer force, with a reasonable cadre of senior officers and ncos with experience in the forces from WW II recruited in Palestine (3 infantry battalions and supporting elements saw service in Italy in 1944-45, after all), as well as others who had served in one or another army in WW II and emigrated to Palestine/Israel afterward, and men with experience in the '48 war and afterwards. If Marcus went as the battalion commander, given his US Army experience, the IDF battalion would actually have better knowledge of US doctrine and practices than any other of the smaller UN contingents; I expect he'd be the only West Pointer among the Allied battalion cos, for example.
Which might affect Israeli doctrine after Korea as well.....