The short answer is that it was allowed to a select few who had the economic means and political permission. Islam as a whole was controlled but not banned in the Soviet Union. The restrictions on Islam relaxed, as did most other limits on free expression, toward the end of the Soviet period. In 1990, the Soviet Union went so far as to charter direct flights between Moscow and Saudi Arabia for the pilgrimage.

This was a stark change from 1945 to 1953, when official pilgrimages were banned, and from the 1920s and 1930s when an atheist campaign violently fought the Islamic presence in central Asia. In 1956, 18 pilgrims were flown from Moscow to Cairo and on to Mecca during what the New York Times called an effort "to counteract the resentment among non-Communist Moslems in the Middle East and Pakistan born of frequent reports about the Soviet government's drive against Mohammedanism within its own borders."

That resentment showed up the year after, when 21 Soviet pilgrims were booed and showered with tomatoes in the streets of Mecca. According to that 1954 account, the Soviet pilgrims were restricted to Mecca proper and were not permitted to participate in the other activities of the Hajj.

The Soviet Union started relaxing restrictions on Islam in the 1950s and 1960s due to the need to maintain good relations with Islamic countries in the Middle East. Muftis from the Soviet Union were de facto diplomats to the Middle East, as Alexandre Bennigsen wrote in an article published in the April 1988 issue of Third World Quarterly.

A 1959 issue of the Times included a fascinating story about how Middle Eastern fans of the Soviet Union were attempting to reconcile communism with Islam. It's not particularly related to the Hajj, but it might be interesting to you. Twenty-five years later, the Times ran a whole series on Islam and the Middle East. Included in that series was an interesting piece about Islam in the Soviet Union in 1974. It says in part:

"In addition, a pilgrimage to Mecca is provided only for a token delegation of 20 to 25 annually. Even the director of the madrasah said he had never been to Mecca. The numbers are small, Sheik Abdullayev said, because without a Saudi embassy in Moscow, pilgrims must go to Egypt or Syria to try to get visas.

But others see it differently. One imam who has been trying for a decade to visit Mecca contended that the pilgrims are chosen by authorities in Moscow and that only 10 to 15 members of each delegation are believers. The rest, he said, are from the K.G.B., the secret police. He said the K.G.B. also had agents in the government's Moslem religious board."

When the direct flights were announced in 1990, the New York Times reported that "Leaders of Islamic organizations and state religious councils here said today that in recent years no more than about 18 or 20 Soviet citizens a year were granted permission by Soviet authorities to make the pilgrimage."

Khadzhi Adil Zeinalov, a leading Islamic figure in Baku, Azerbaijan, was one of only three people permitted to go on Hajj from Azerbaijan in 1982, for example. When the direct flights were announced, he explained that it took months to navigate the bureaucracy needed for the trip's approval.

Under Soviet law, children could receive religious training only at home. At school and in public, atheism was the lesson of the day. The Soviet government operated directorates that administered Islamic practice. It named officials to lead those directorates. In practice, these officials executed the orders of Moscow and were used in campaigns against such things as pilgrimages, even local ones within the bounds of the Soviet Union.

Even after the Soviet Union, it took time for the Islamic regions of the former Soviet states to pick up participation. In 2000, for example, only about 520 people went on Hajj from Tajikistan. By 2008, the figure was 10 times that. An improving economy, the end of a civil war, and the Islamization of the region were two factors in the increase.

Adeeb Khalid's Islam after Communism: Religion and Politics in Central Asia was published in 2007 by University of California Press and would be a good read for you, particularly the third through fifth chapters, which cover Islam between 1917 and 1991 in the Soviet Union. Khalid's written some pretty good papers as well, which you can find if you search JSTOR or some other database.

I also suggest picking up Martha Brill Olcott's In the Whirlwind of Jihad, which examines Islam in Uzbekistan. But Olcott's book won't be a precise answer to your question. The issue of Islam in the Soviet Union needs to be explored more, as she herself states in the fourth chapter: "The task of writing a complete history of Islam in this period will rest with another generation, as it requires that the archives of the Spiritual Administration of the Muslims of Central Asia and Kazakhstan as well as Soviet archives be exhaustively researched, preferably by a scholar with little stake in the outcome of the findings."