"we didn't show cartoons from other countries that weren't close allies like the UK"

As someone who grew up watching British Cosgrove Hall series in the 1980s like Danger Mouse, I strongly question that last premise! Danger Mouse was actually the first animated program to play on the US cable channel Nickelodeon in 1984, and the channel relied pretty heavily on imported animation for the rest of the decade. But to move to Russia, I think the time periods need a bit of straightening out. There was a period where relations were "meh" - this being the Cold War, and during that time it is correct that you wouldn't see any American cartoons broadcast over Soviet airwaves or screened in movie theaters.

This also overlaps with the height of the Soviet Union's own animation industry, which at the time was one of the largest in the world. It was organized into one state-owned studio known as Soyuzmultfilm ("Union Cartoon") in 1936, and this studio produced an impressive output, especially in the 1960s through 1980s. If I were to pick two particular animated series as representative of this period, in part because of just how universally-known they still are in the former Soviet Union, I'd have to go with Nu, pogodi! ("Well, Just You Wait!"), which was first produced in 1969 (and despite a big slow down after 1993 still gets an episode made every now and then) and Boris Zakhoder's Winnie-the-Pooh, based off of the A.A. Milne book and not the Disney animations, which was first made the same year. These are just two examples, and the output was incredibly wide and varied over this period - it was the largest animation studio in Europe, with, interestingly enough, one of the other biggest studios at the time being Pannonia Film Studio in communist Hungary.

By the 1980s, Soyuzmultfilm was pursuing joint projects with other countries, and by the late 1980s in particular this included partners from such countries as Japan and the UK. A prominent example of the former is the 1986 Adventures of Lolo the Penguin, directed by Gennady Sokolsky and Kenji Yoshida, and jointly produced by Soyuzmultfilm and Japan's Lifework and Enoki Films.

Much of this came about because of Gorbachev's perestroika and glasnost programs, which began to pick up steam around 1987. The USSR tried to seek more foreign partners for all sorts of trade relationships as part of its goal of economic restructuring - this was also the period where there were major reductions in tensions from the Cold War, with the Cold War proper being declared officially over in December 1989.

This economic restructuring impacted Soyuzmultfilm itself. In the late 1980s, a number of its projects were reorganized into "creative production associations". Big changes came around 1989 when the studio became a "leased enterprise" - the animators themselves had to lease the facilities from the state. On top of this, because of the increasingly bad economic situation starting in 1989, the Soviet State Committee for Cinematography, which had overseen Soyuzmultfilm since its founding, seriously cut back funding for the studio (the State Committee itself would be abolished by the end of 1991). These events, coupled with animators leaving the studio, seriously weakened the studio and drastically reduced its output.

Now all of this came at a time when cultural imports from the United States became very accessible to the Soviet public. The first evidence of US animation that I can find is of Disney's Snow White and the Seven Dwarves being screened to a select audience in Moscow on the film's 50th anniversary in 1987. President Reagan would visit Moscow in May 1988, but he would only be paving the way for Mickey Mouse's visit (with 45 Disney executives) on October 15, 1988. Mickey was met at Sheremetyevo Airport by Misha the Bear (the Moscow 1980 Olympics mascot, designed by Soyuzmultfilm), and was there to kick off a two-week film festival of Disney animation shown in Moscow, Leningrad and Tallinn as part of his 60th anniversary. The films shown included Fantasia (whose premier was attended by Raisa Gorbacheva and 100 orphans), Bambi, Snow White and 101 Dalmations, as well as a number of animated shorts. Roy Disney himself was at the 101 Dalmations premier and even fielded questions from the audience and journalists - he said that a Soviet Disneyland was of interest but not imminent, as the company was focusing on opening Euro Disneyland (which opened in 1992), but planned to add a Soviet pavilion to Epcot Center. Disney noted the company would be interested in screening more animation in the USSR, but that box office receipts shares and protection of intellectual property had to be negotiated first (I'm not sure it was a particular dig at the Soviet Winnie the Pooh, or the Soviet Mary Poppins, but they'd at least have been in the back of his mind).

In any case, it seems that eventually negotiations worked out, as 1991 saw the broadcast of the first US animation on Soviet tv, and it was Disney's Duck Tales, followed by Chip n' Dale: Rescue Rangers. This was part of a big inrush of US cultural products, ranging from Schwarzenegger films to MTV videos to soap operas like Dallas. In any event, the USSR itself dissolved by the end of the year, and privatizations, economic chaos and lifting of import restrictions would mean that a great many foreign cartoons (as well as other tv shows and films) would rush into the former Soviet market in the 1990s.