Really interesting question! The White Album actually caused quite a bit of a stir when it came out as Back In The USSR was not the only politically charged song - in fact, all 3 of the songwriters in the group added something to the double LP that the media latched on to: Paul with USSR, John with Revolution and George with Piggies. It just seemed to be on their mind, especially considering they had just spent a considerable amount of time away from politics and international relations in an ashram in India.

As you brought up, Back In The USSR was a kind of mashup parody - McCartney was lampooning Chuck Berry's overtly nationalistic "Back In The USA" as well as Beach Boys-esque harmonies. I'll start with the Beach Boys, as this was actually not the first time the group had poked fun at their American counterparts; "Girl" from Rubber Soul featured Paul and George singing "tit tit tit" against John's bridge in a style emulating The Beach Boys but drawing attention to their 'good boy' image. However this wasn't an aggressive dig, just lighthearted fun, and Back In The USSR was intended in the same way. In fact, Mike Love was with The Beatles in India and (supposedly, according to Mike Love) was the one who suggested the bridge should be like California Girls! His autobiography looks back at the song fondly (he also has claimed to have written other chunks of White Album songs and "jokingly" asked for royalities), so we can infer that if the most notoriously grumpy member of The Beach Boys was okay with it, the rest of the band must not have minded! Chuck Berry and other early rock n rollers were going out of fashion by the late 1960s, so poking fun at their style and vocal delivery was inline with The Beatles' humour - especially one so American as Chuck Berry

The media reaction is more difficult to generalise - McCartney was banned from performing in the USSR for a period of time, but so were all of The Beatles and a heap of other artists so its difficult to tell how much of it was down to one specific song. We do know that Elton John was asked not to sing it amongst others (he did it anyway), but again the USSR was a big and complicated media machine. On home soil, a great deal of music reviewers and journalists recognised it for what it was - a parody. Other songs on the album like Piggies were much stronger in their critique of capitalism and the upper classes. Back In The USSR was the album opener, however, and some Beatles historians have looked back on this as a distinct choice to take the listener by surprise - recognisably American music backed with the most un-American lyrics imaginable.

What we have to remember with the Soviet reaction is that the government and media didn't really like The Beatles in the first place. They were a bastion of Western imagery and a pro-USSR song wasn't likely to make a difference! Its very easy to hype up USSR reactions or American outrage after-the-fact, but what i've gathered from reading too many Beatles books and musciology articles is that it was a joke, and most people laughed.

Edit: Grammar

In the U.S. and U.K. at least, "Back In the U.S.S.R." was widely understood upon its release to be a satire, a deliberate pastiche of Chuck Berry and Beach Boys style late 1950s/early 60s Americana ironically looking fondly at the Soviet Commies.

This is found in contemporary reviews such as in Rolling Stone, American Eagle, Record Mirror, and the New York Times. As an example, from the 1968 Rolling Stone review:

"...the Beatles can safely afford to be eclectic, deliberately borrowing and accepting any outside influence or idea or emotion, because their own musical ability and personal/spiritual/artistic identity is so strong that they make it uniquely theirs, and uniquely the Beatles. They are so good that they not only expand the idiom, but they are also able to penetrate it and take it further.

"“Back in the USSR,” this album’s first track, is, of course, a perfect example of all this: it is not just an imitation (only in parts) of the Beach Boys, but an imitation of the Beach Boys imitating Chuck Berry. This is hardly an original concept or thing to do: just in the past few months we have been deluged with talk of “going back to rock and roll,” so much that the idea (first expressed in the pages of Rolling Stone) is now a tiresome one, because it is, like all other superficial changes in rock and roll styles, one that soon becomes faddish, over-used and tired-out."

The review hints at the nostalgic trend at the time just emerging to look back at the 1950s, which would later be typified by bands like Sha-Na-Na, and then in the 1970s, in media like American Graffiti, Grease, and Happy Days. In 1969, John Lennon played at a concert festival called the "Toronto Rock & Roll Revival" on the same bill as Chuck Berry, Little Richard, Bo Diddley, Jerry Lee Lewis, and others. They played alongside contemporary acts such as Lennon's Plastic Ono Band, Alice Cooper, and The Doors. (The Plastic Ono Band set would be released as the live album Live Peace In Toronto in late 1969.) The 1950s "rock and roll revival", a reaction to the psychedelic and early prog-rock scenes, accused by some as being too self-important, was thus already being recognized by the end of 1968. The contemporary reviews tended to look at "Back In the U.S.S.R." more from that standpoint, rather than from a political point of view.

On mobile, sorry if there are spelling errors.