It's likely that the Soviet Union did not experience a baby boom to replace the demographic deficit it suffered from the war. The country's situation was very different from those in "the West", particularly the United States and United Kingdom. While the latter two largely had their civilian populations shielded from the genocidal efforts of the Nazis, the Soviet Union bore the full brunt of an intense and murderous war which took the lives of 26-27 million people. 20 million of these deaths, or 76%, were males, leading to an incredible gender imbalance. Overall there were 96.6 million females and 74.4 million males. In the age range of the majority of the childbearing population, 20-44, there were 37.6 million women and 27.6 million men. In the countryside this ratio worsened to 27.6 men for every 100 women in all rural areas - some regions such as Smolensk had a ratio as low as 19 men for every 100 women.

Further, we need to include demographic losses - children who would have been born from 1941-45 if war hadn't occurred, based on prewar fertility trends. Between 1940 and 1943 the number of births per 1,000 citizens declined from 35.3 babies to 11.2, a precipitous drop. In total, demographic losses are estimated at around 9.1 million children, accounting for the fact that Soviet fertility was already declining throughout the 1930s. In total, then, the Soviet Union ended the war with a deficit of 35-36 million people, by the official count (With other estimates ranging even higher).

Such as destructive war further damaged the fertility of the population in other ways. Millions of people were displaced by the war, including perhaps 16.5 million who fled from the Germans or were evacuated by the Soviet government and 3.1 million Soviet citizens who were used as Ostarbeiter slave laborers in Germany. The Soviet regime further exacerbated this displacement by deporting 1.9 million ethnic minorities suspected of disloyalty. These mass population movement seperated families and worsened living conditions, greatly decreasing fertility.

As mentioned above, demographic disparities in the countryside were severe when the war ended. Poor living conditions in the countryside exacerbated these woes - as Mark Edele describes it, veterans returned home after demobilization, took a look around, and left as soon as they could. As a result, even in early 1946 there were only 6.5 million able-bodied young men in the countryside. Even as late as 1956, for example, of 1,844 families surveyed as living in dugouts, 1,440 of them were in rural areas. The result is that gender disparities continued for long periods of time, reducing opportunities for children across the board.

Standards of living also remained terrible well into the 1950s, making women less likely to have children, more likely to get an illegal abortion, and the children themselves more likely to die young. In 1956, for example, a study of communities across the Soviet Union found that almost 40,000 families continued to live in structures unsafe for human habitation (Barns, railway cars, etc). Overall housing stock, despite doubling between 1945 and 1950 to 513 million square meters (From 280 million in 1940), actually fell from 5 square meters per person to 4.67 because of massive in-migration to cities. Even in Moscow, half the housing stock didn't have sewage, clean water, or central heating. In rural areas and provincial cities this problem only worsened - in 1951 35% of dwellings in the city of Molotov lacked clean water. Most Soviet citizens lived in abject poverty well into the 1950s.

A number of state actions also worked to lower fertility. Aside from deportations, the government condemned millions to forced labor, with the number peaking at 2,472,247 in 1953. Further, the majority of prisoners received sentences longer than 5 years, with 31% serving sentences longer than 10 years. Harsher laws on "shirking" from work and other economic crimes were the main reason for this increase. Millions of individuals were cut out of the economy and couldn't have children.

According to Mie Nakachi, the foremost historian on postwar reproduction, the famine of 1946-47 caused the single greatest drop in fertility. While the effects of the Second World War and drought certainly caused initial drops in crop production, the state's unrealistic grain quotas, withholding of workers wages, and removal of many individuals and dependents from rationing led to the starvation of over a million people and malnutrition for millions more. The majority of deaths of infants from 1946-47 was caused by malnutrition or abandonment because the family was starving. In contrast to the increase from 2.5 million to 4 million from 1945-46, birth rates only increased to 4.5 million in 1947 and declined to 4.2 million in 1948, leading to panic within the Soviet government - rather than addressing the root causes, famine and poor living conditions, they chose to blame abortions.

To sum it up, the Soviet Union's "baby boom" was smothered in the cradle. While it experienced initially explosive growth from 1945-46, numbers stagnated afterwards. Following the famine birth rates were 5 million in 1949, 4.8 million in 1950, 4.9 million in 1951, and 5 million again in 1952. The Soviet population only recovered to its prewar levels by 1954. No true "baby boom" ever occurred. I think Mark Harrison makes the best summary of the postwar demographic situation:

Therefore, depending on one's assumptions about postwar fertility rates in the absence of World War II, the postwar demographic deficit was either simply not made up, or continued to increase indefinitely throughout the postwar period

Sources:

Alexopoulos, Golfo, "Amnesty 1945: The Revolving Door of Stalin's Gulag"

Edele, Mark, "Veterans and the Village: The Impact of Red Army Demobilization on Soviet Urbanization, 1945–1955"

Ellman, Michael, and Maksudov, S., "Soviet Deaths in the Great Patriotic War: A Note"

Filtzer, Donald, "Standard of Living versus Quality of Life: Struggling with the Urban Environment in Russia During the Early Years of Post-War Reconstruction" in Late Stalinist Russia: Society Between Reconstruction and Reinvention

Ganson, Nicholas, "The Soviet Famine of 1946-47 in Global and Historical Perspective"

Grinchenko, Galinada, "The Ostarbeiter of Nazi Germany in Soviet and Post-Soviet Ukrainian Historical Memory "

Harrison, Mark, "Trends in Soviet labour productivity, 1928-85: War, postwar recovery, and slowdown"

Manley, Rebecca, To the Tashkent Station: Evacuation and Survival in the Soviet Union at War

Nakachi, Mie "Replacing the Dead: the Politics of Reproduction in the Postwar Soviet Union, 1944-1955 " (PhD diss)

-- "Population, Politics and Reproduction: Late Stalinism and its Legacy" in Late Stalinist Russia: Society Between Reconstruction and Reinvention

Zubkova, Elena, Russia After the War: Hopes, Illusions and Disappointments, 1945-1957

The UK did (after both World Wars). See this graph from the government's Office for National Statistics.