How well-known were Tolkien's works in the USSR or the Eastern Bloc in general?

Hi! To give at least a provisionary answer to a question that might as well sink unanswered forever, I wanted to point out that yes, Tolkien's works did make their way to USSR, albeit late — starting a rather sizeable following that partly overlapped with local hippies, and spawning one of the first and largest fandoms in the region, that has largely introduced things like fan fiction, LARP, conventions, cosplay, HEMA, and historical reconstruction to local audiences. Note that my answer is limited to USSR and mostly focused on the Russian Republic.

Here is a bibliography study by the main Russian Tolkien fandom website, established back in 1996. I will lean on it, since it appears incredibly meticulous and thoroughly sourced (down to minute mentions or dictionary appearances).

It says that an excerpt of Chapter 7 of The Hobbit was translated as early as 1969, in an "international cultural exchange" magazine Anglia (England). Then, during the 1970s, Tolkien gets briefly mentioned in textbooks or critical overviews of foreign speculative fiction, until in 1976 The Hobbit (being, in essence, a rather inoffensive fairly tale with ethnographic flavor) is translated and published. The print run is a very modest (FOR USSR) 100 000 copies.

(As an aside: I was born in 1984 and read The Hobbit as a small kid, in an illustrated, dog-eared, Soviet edition in this translation).

Over the next few years, Tolkien and LotR are mentioned benignly in lit-crit articles and papers, and finally, in 1982, the first book is translated and published as Keepers (Hraniteli, Хранители), a slightly abridged version of the Fellowship. This translation still has its die-hard defenders. There is no controversy in contemporary reviews: the stature of Tolkien as a modern classic abroad is well known and accepted (if tempered by the traditional slightly dismissive attitude to both genre fiction and children's lit).

During the 1980s, as Hraniteli is being translated from Russian to Armenian, Lithuanian, Ukrainian, and Moldovan, we're (officially) still waiting for the whole thing. In 1988, the unabridged translation of the same first book appears. Also, there is a smattering of extremely low-budget TV teleplays adapting Hobbit or FotR to screen. All in all, USSR never saw the entire Lord of the Rings trilogy published officially...

...apart from "That Book", an infamous 1990 abridged version, that condensed all three volumes into one paperback while severely butchering the plot, inventing additional storylines, and retelling all in fairy-tale speak. Much later, it turned out that it was a super-early, 1966 attempt at translation, designed by the translator to fly under the radar of censorship and be published as a run-of-the-mill children's sci-fi — even introducing an umbrella sci-fi plot about scientists studying the alien device (The Ring) and downloading the information about "the past" from it. It ultimately wasn't picked up for publishing because it just seemed too weird, and it was unclear who its audience was. Then, 20 years later, it was published without the sci-fi bits, causing confusion and wrath among hardcore Tolkien fans.

Nevertheless, Soviet readers were treated to over 400 000 copies of Fellowship of the Ring and 500 000 copies of The Hobbit in total.

But very importantly, this does not take into account the unofficial translations and prints of the books. And LotR was one of the more popular samizdat pieces (xeroxed or typed copies), comparable among the intelligentsia to Doctor Zhivago, Hemingway, and Solzhenitsyn.

As such, Soviet literary circles, countercultural groups, and The System (a rather unintuitive name for the Soviet hippie network) were well aware what happened to Frodo at Mount Doom. And Tolkienists (even RPing as Tolkien's races and characters), were known to some in the Soviet Union since late 1970s.

Towards the late 1980s, "Tolkienism" got a new spurt of life in the liberalizing USSR; the first LARPers appeared, and they were all devoted Tolkienists. In fact, the word became firmly welded to the concept of LARP (and cons, and cosplay) in Russia for the next 15 years. The very first large-scale Tolkienist Hobbit Revelries LARP was conducted still in USSR, in 1990.

This absolutely raging subculture continued strong into the tumultous and lean 1990s, the austere adventurer cosplay and lifestyle a great fit for poor Russian high schoolers and university students (as well as Russian suburban forests and city parks being a good fit for LARPing Middle Earth. And drinking). If anything in late USSR/early Russia could equal the entire nerd/con culture specific to US, it was this (plus, Russian historical reconstruction people obviously grew out more or less from this fandom, too).

Unsurprisingly, LotR (and Silmarillion — which had 4 translations in 1992-1993 alone) was never out of print since and received dozens of translations, one of which I've read as a middleschooler in the 90s (it had a mind-bendingly good set of color illuminations by an old-school Soviet illustrator, mixing Medieval art, Orthodox icons, and surrealism). Later, having spawned the Russian geek culture, Tolkienism gave way to the countless fandoms active in the country now, and is way less prominent compared to its former ubiquity.

As far as I know, Soviet/Russian fandom also hatched a large culture of fan fiction, gradually transitioning into numerous unlicensed, commercially published extended universe books, which took itself as seriously as Tolkienists took Tolkien (which is almost like God Almighty and his scriptures — they actually call it Apocrypha in Russian tolkienist circles). At least one of the authors, Nick Perumov, went on to become one of the best selling and prolific fantasy writers in Russia (or, rather, out of Russia: he works a microbiologist in US since 1998 along with his literary career).