This is a pretty complicated question, mostly because the ideological bent of pan-Arab or Arab nationalist movements changed pretty dramatically over time.

The TL;DR is probably something like: pan-Arabism didn't start as either socialist or anti-Western, to understand why it became both look to the inter-war years of anti-colonialism and to the Suez Crisis.

So to begin, and apologies for the length:

Early groups operating in the 1910s and 20's like al-Fatat and al-Ahd were composed largely of societal elites like Arab Ottoman military officers or the intelligentsia. The origins of these movements came in the form of cultural resistance to Turkification polices of the CUP (the "Young Turks") but over time and by the outbreak of the First World War there was a growing feeling cultural equality was not sufficient and that national independence was the solution. The Arab Revolt has been described as the culmination of that feeling, but there's some push back on that insofar as the revolt emerged and had it's greatest support from the bedouins of Arabia, rather than the peoples where pre-war Arab nationalist sentiment had been greatest in Greater Syria or Iraq. Nonetheless, the Arab Revolt also had a strong clique of Arab former-Ottoman officers and members of both al-Fatat and al-Ahad. Faisal himself joined al-Fatat in at least cursory form before the revolt broke out.

From the Hussein McMahon correspondences, the objectives of the revolt were pan-Arab (at least, for the former Ottoman territories, there was no discussion at this point of uniting Egypt with any of these lands. It's an interesting side point to discuss at which point the Egyptians actually started to view themselves entirely as "Arabs" but I haven't read enough on that issue to fully dive into that. Suffice it to say that it's my understanding that they would have viewed themselves as such by this time period around WWI). They were also Hashemite monarchist (at least in Syria and Arabia, not nearly as much in Iraq) not socialists. We also know from the King-Crane commission after the war that the broad community of Syria was not actually anti-Western as such. Recognizing that some sort of international western supervision was necessary following the war they actually requested American oversight, and barring that British. The peoples of Greater Syria were most adamantly anti-French. Unfortunately the French is who they got.

It's in this period of imperial control in the inter-war years that you can start to see an anti-colonial, or even anti-western, streak in pan-Arab thought emerge. The Muslim Brotherhood is formed in Egypt in 1928 and obviously prevents a radically different kind of national identity than other such organizations had done in the past. They were also markedly and explicitly religious, identifying arguably as much as Muslims as they did as Arabs. The MB soon had branches in Transjordan, Palestine, and Syria, though they were dwarfed in size and importance by the original Egyptian branch.

It was in these inter-war years that radical politics in the Arab world, just as in Europe, started to become more mainstream. The various "colored shirt" street gangs of the fascists, communists, Wafdists, MB and other groups battled it out on the streets of Cairo just as similar groups battled in the streets of Germany.

Britain, in particular, tried to placate its colonies by granting a sort of pseudo-independence. Britain issued a "unilateral declaration of independence" on behalf of Egypt that reserved foreign policy issues, the Sudan and the Suez Canal to itself. It granted Iraq independence in 1932. These did little to stifle nationalist movements, and its worth noting that in Egypt, to this day, "28 February 1922" is not a public holiday or commemorated in any way because it is recognized, rightly so, as a sham.

Nonetheless, in the inter-war years I think you could argue that, if anything, it was fascism, not bolshevism, that got the most attention and support, particularly as it became clear that Germany and the Axis were going to be opposed to Britain and France. As a result when the war broke out Egypt tried to maintain its neutrality, or even a pro-Axis policies, until the Abdeen Palace Incident when the British military basically all but carried out a coup and basically left King Farouk a broken man, even if he was still titularly king. A similar story played out in Iraq in 1941. The result of these was a brewing post-war rejection of Monarchial nationalism. In Syria and Lebanon I think you could make the case that the result of that pro-Axis feeling was that these territories were more pro-Vichy than Vichy itself and the allied invasion of Syria saw some pretty intense resistance.

The immediate post-war period was interesting because France, having just fought a war against Fascist occupation, was now in the awkward position of having to make good on its promises and grant Syria and Lebanon independence.

But it was the Israeli independence war that really made revolutionary pan-Arab nationalism a front and center issue. The ideology was already in place. Michel Aflaq had founded the Baath Party in Syria in 1945. The Muslim Brotherhood was in contact with officers in the Egyptian army about a possible coup in these immediate post-war years. Many of those officers would form the core of the Free Officers Movement. But it was the disastrous performance of the individual national Arab armies in the war against Israel that really sparked a renewed push for a fully-fledged 20th century pan-Arab nationalism in Egypt, Syria and Iraq.

Syria suffered a succession of coups following the '48 war, and the Egyptian free-officers took control in 1952.

Phew. Ok. that gets us pan-Arabism, but why socialism? As I think I've made clear, pan Arabism before this point might easily have gone in other directions. It was originally elitist, then monarchist, then anti-colonial and anti-monarchist, then militarist with the free officers. So how did become anti-western and usually pro-soviet?

The Suez Crisis of 1956 was the real pivotal moment. Before this, the US thought that Nasser could be a critical ally in the region, and even play a role in the Baghdad Pact alliance which they tried to put together in 1955 as a kind of Middle Eastern version of NATO.

Nasser wanted a few things himself. He wanted to circumvent the regional arms embargo in place since 1948 (after all, the Israelis were getting covert support from the French.) He wanted to build the Aswan Dam and electrify Egypt. And he wanted control of the Suez, the last bastion of Colonialism in Egypt.

When Nasser brokered an arms deal through Hungary, the US withdrew its funding for Aswan. Nasser saw then thought that the only way to get that money would be to nationalize the canal, which he did in 1956. Moreover, the soviets promised to step in to provide technical support. The British, the French and the Israelis then concocted an incredible conspiracy where the Israelis would launch a false-flag invasion of the Sinai and the British and French would swoop in as "peace keepers" and restore their control of the canal. The conspiracy outed itself soon after the war broke out and the US was furious, stopping the war by threatening to dump sterling.

This was the height of Nasser's reputation, and these machinations now put him pretty firmly in the "socialist" side of pan-Arab Socialism. The rest of the Arab world, under the glow of Nasserism, followed suit, and the character of most other pan-Arab movements followed his lead through the 50's and 60's. The success of the socialist Baath Party in Syria, where they were the second largest party in the country and which was founded by Michel Aflaq in 1945, propelled the creation of the United Arab Republic in 1958, though it would only last 3 years.

Aside from these great-power Cold War struggles, socialism also meant secularism. Nasser, shortly after taking power, was engaged in an incredibly violent struggle against the Muslim Brotherhood. Michel Aflaq, like many Syrian Christians, supported pan-Arab Socialism partly as an alternative pan-Islamism. Saddam Hussein, perhaps the most famous Baathist, was a student in Cairo during these critical years at the height of Nasserism from 1959-1963.

Though the United Arab Republic would break up in 1961, the flavor and ideals of the movement were set. Most of these countries were not technically part of the soviet sphere, but were rather part of the non-Aligned movement. Still, under the regional leadership of Nasser, the Arab nationalist parties were certainly more aligned with the Soviets than with the US.

I think that's a good place to stop, but in terms of when this began to change, the critical year was 1979 which saw peace between Egypt and Israel, the Iranian revolution, rise to power of Saddam Hussein, and subsequent Iran-Iraq war, the Soviet invasion of Afghanistan and the '79 Oil Crisis.

Source wise:

phew, we've gone a bit all over the place. The best one volume history of this that I always point to is Eugene Rogan's "The Arabs: A History."

For some of the more in-depth issues, on the Muslim Brotherhood please see Mitchell's The Society of Muslim Brothers which also has much context on inter-war Egypt and the MB's connections with the free officers.

For Egypt generally, see the last revised edition of P.J. Vatikiotis' History of Modern Egypt: From Muhammad Ali to Mubarak.

On the Arab revolt, the Suez Crisis and Imperialism in the region check out Western Imperialism in the Middle East by D.K. Fieldhouse.

For more on Iraq the relevant chapters of Charles Tripp's A History of Iraq.

I have to admit that Syria and Lebanon are week points of mine, as is the Free Officers Movement in Libya under Gaddafi. Both topics are covered in Rogan's book, but I'm afraid I can't point to anything deeper and I apologize for not touching on Qaddafi.