More can definitely be said, but I've written before about the development of pro-natalist policies in the USSR bracketing the war years, which is definitely relevant here, so I'll repost the older answer here.

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Looking at natalist policies in the Soviet Union, especially with regards to abortion, we can see a lot of policy being driven by concerns about the birthrate, and its rise and fall. Especially at the time of the war, there was very explicit concerns raised about the issue and policies were changed and created with the explicit goal of raising it.

In the Russian Empire, and the first few years of Bolshevik rule in Russia, abortion was illegal. But, as in most places where the procedure is illegal though, the procedure was nevertheless popular, but insanely dangerous. One observer pre-1920 noted:

Within the past six months, among 100 to 150 young people under age 25, I have seen 15 to 20 percent of them making abortions without a doctor's help. They simply use household products: They drink bleach and other poisonous mixtures.

The decision to legalize the procedure, and make it simple to obtain, was almost entirely a practical decision. In 1920 they became legal if done by a doctor, essentially in acknowledgement that it would happen no matter what, so the state should do its best to make it safe. They were subsidized by the state, so free to the woman. In 1926, the abortion rate was 42.8 per 1000 working women, and 45.2 per 1000 'housewives' (compare to the US today, at 13.2 per 1000 women. Modern Russia continues to be very high, at 37.4 per 1000 or so)

But this wasn't to remain. As noted, the change was not because abortion was seen as good, but that legalizing it was a necessary evil and that the state would work to eliminate the underlying economic reasons driving women to have them. As it turned out, poor women were no more likely to be using this 'service though'. If anything, it was the better off women who were getting more abortions. Even worse, the birthrate in the USSR was falling precipitously, from 42.2 per 1000 in 1928 to 31.0 in 1932, according to a government study released in 1934. Thus the law changed in 1936 when policies started to return to pushing more 'traditional' gender roles for women, and included restricting abortion again - it required a medical reason now. As before though, just because it is illegal doesn't mean women don't seek them. After 1936, "back-alley" abortions were on the rise, and they certainly carried additional risks with them, and penalties for obtaining one meant injured women would only be further harmed by not seeking treatment:

Women who became infected during these procedures or who sought assistance for heavy bleeding were often interrogated at the hospital before they were treated, as the authorities attempted to learn the names of underground abortionists. Abortionists were punished with one or two years’ imprisonment if they were physicians and at least three if they were not. The woman herself received a reprimand for her first offense and a fine if caught again.

Abortion statistics aren't readily available for this period, but my book notes that as the birth rate didn't seem to change much - rising briefly through 1937 when it reached 39.6 per 1000 but again beginning to decline until leveling out at 33.6 per 1000 in 1940, the same rate as 1936 when the law went into effect - as the laws became restrictive again, this would imply women weren't especially deterred by the law and continued to seek them at the same rate as before (see 1926 numbers), if not higher. There was no ready access to, nor education regarding, other means of birth control (Aside from abortion as birth control, by far most common being 'coitus interruptus'), so it was really the only means of family planning available to women.

The massive population losses that occurred in the early 1940s further increased pro-natal policy planning, but with both carrots and sticks. Laws to assist so called "war widows" (referring not simply to women who lost husbands, but women who lost the potential for a husband due to the decline in the male population) both in raising their children as single mothers as well as having children in the first place.

Soviet propaganda campaigns to encourage motherhood predated the war even, but the massive calamity of course kicked it into overdrive. During the war, there was a definite decline in the birthrate due to "general decline in the reproductive health of mothers, as reflected in the high rate of premature births", as characterized by the People’s Commissar of Public Health G.A. Miterev, and Soviet leadership worked hard to try to turn that around, with their clear awareness that to see further decline would imperil the ability of the USSR to bounce back in the long term.

Programs and incentives to encourage motherhood existed, such as awards for bearing a certain number of children and various state assistance programs for both married single mothers, while legal penalties were either added or increased, most especially with the Family Law of 1944, which further penalized abortion and increasingly penalized divorce as well. The shortage of men also meant a very important shift, in which the Soviets worked to try and both destigmatize single-motherhood by increasing state benefits they could receive and featuring mothers of ambiguous marital status in propaganda, while also tacitly encourage even married men to sleep around by preventing the single mothers from suing the father for child support, and making it harder for their irate wives to divorce them. The result being that many men would have numerous affairs, and even unmarried men would often bounce from relationship to relationship.

Now as to your question, which is basically whether or not the Soviets were successful in reversing the trend during the war years? Well, not terribly. There was a definite boost in the fertility rate immediately after the war years, but it was rather short lived, and quickly began to decline again. Here is a table of the fertility rates of the US and USSR, which allows for a comparison of the 'Baby Boom' in America, for the period in question:

Year	USA Total Fertility	USSR Total Fertility	-	Year	USA Total Fertility	USSR Total Fertility
1926	2,909	5,566	-	1944	2,567	1,942
1927	2,827	5,418	-	1945	2,491	1,762
1928	2,656	5,318	-	1946	2,942	2,868
1929	2,524	4,985	-	1947	3,273	3,232
1930	2,508	4,826	-	1948	3,108	3,079
1931	2,376	4,255	-	1949	3,110	3,007
1932	2,288	3,573	-	1950	3,090	2,851
1933	2,147	3,621	-	1951	3,268	2,914
1934	2,204	2,904	-	1952	3,357	2,898
1935	2,163	3,263	-	1954	3,541	2,974
1936	2,119	3,652	-	1955	3,578	2,909
1937	2,147	4,308	-	1956	3,688	2,899
1938	2,199	4,351	-	1957	3,767	2,903
1939	2,154	3,964	-	1958	3,703	2,940
1940	2,301	3,752	-	1959	3,712	2,903
1941	2,399	3,742	-	1960	3,653	2,940
1942	2,628	2,933	-	1961	3,627	2,879
1943	2,718	2,366	-	1962	3,471	2,755

So as you can see, they did bounce, with a sharp - and important - increase in 1946 and 1947, but certainly didn't regain pre-war levels like we see in the US, and even bigger, while they had been far higher than the US before the war, the total fertility rate is now noticeably lower (with a minor exception being, when broken into age cohorts, a higher rate in the USSR for women over 30) and stabilized much quicker within a few years of the war (stabilized being a relative term. there would be later drops). So all in all, yes, there was a brief boom that we can see, and it likely was quite important as far as the stability of Soviet population numbers go, but it wasn't as long lasting as we see in the US, puttering out somewhat quickly.

Edit: The table in graphic form courtesy of /u/iamjoesusername.

A decade later, Soviet society became one of the world's most socially conservative.

The most sexually conservative? Maybe compared to the most liberal places in the USA in the 80s, but definitely not in the world. In general, sex was not somehow prohibited (except for homosexual contacts, but these laws were rarely observed), but culturally it was simply not discussed. It's like we know that there is sex and you can do it, but silently lol

Or did it remain a first wave feminist society for the rest of its existence after the 1920s?

It was the most feminist society in the world at that time tho