Soviet Union in that time?
As a quick aside, we would know that the Soviet Union did not use any forms of encryption at this time as the Soviet Union did not exist yet. The ratification of the USSR occurred in December 1922 (a year after the conclusion of the war at the Treaty of Riga in '21). Bolshevik or Soviet Russia would be the appropriate party to the conflict. I am sure that's whom you meant but it is worth noting all the same.
Some more digging around seems to indicate the events you are referring to are Miracle of Ciechanów when the 203rd Uhlans sezied a Russian radio station and also refers to the Sekcja Szyfrów, the precursor the Polish Cipher Bureau.
A team of mathematicians, students and cryptologists managed to decipher messages showing a gap in Soviet lines which the Poles used to gain victory.
Well we can infer a couple of things regarding Russian cryptography in the early 20th century (the inter-war period.) The Austro-Hungarians were able to break the Russians version of a Viginere Cipher (A poly-alphabetic substitution cipher) during the First World War to great effect.
What Wikipedia does tell us about Russia's level of encryption sophistication during World War One is that they were at least capable of developing poly-alphabetic ciphers.
During World War I, Pokorny, as a cryptologist in the rank of major,
headed the Austro-Hungarian General Staff's Russian-Cipher Bureau. He
showed great ability in decrypting Russian enciphered military
messages that were broadcast by radio in 1914–17. He recognized that
the Russian cryptographers had reduced the 35-letter Russian alphabet
to 24 letters, while doubling the 11 missing letters to some of the
other 24 letters
So what we know that the schemas in Soviet Russia at the time would be some variants of both poly-alphabetic (e.g Viginere Ciphers) and mono-alphabetic substitution ciphers; as well as codewords which are not ciphers per se, but are forms of encryption.
However it is also important to realise that historically commanders may have used simpler ciphers or codes at the local level in order to avoid transmitting in the clear. Whilst the Soviets may have possessed more sophisticated ciphers than a simple Caesar shift (a monoalphabetic substitution ) - they may not have had time, inclination or expertise to encrypt a message fully.
Encrypting a message poorly can result in the breaking of an entire cryptographic system - by exposing the underlying logic used to cipher the message. It was quite common in World War One for soldiers to send a message in the clear rather than encrypt them if they did not have the time to do so.
The British and French were already familiar with such problems in
"communications discipline". They hadn't completely solved the
problems either, but they had at least managed to get it through the
heads of most of their signalmen that if they didn't have time to
properly encrypt a message, they shouldn't bother trying; send the
message unencrypted, or "in the clear". A partially or badly encrypted
message could undermine a cipher or code system, sometimes completely,
which made an unencrypted message far preferable.
So it's quite likely that Caesar ciphers could have been at use despite the possession of more sophisticated ciphers at the time for reasons of expediency. However the reasons seem to be much more familiar than that:
Russian army staffs were still following the same disastrously ill-disciplined signals-security procedures as had Tsarist army staffs during World War I...
So what we are able to tell from this source that the Russians used over 100 different ciphers, some of which we would can reliably infer would be poly-alphabetic, but in also many events either transmitted in the clear or:
encrypted by means of such an incredibly uncomplicated system that for our trained specialists reading the messages was child's play.
In the crucial month of August 1920 alone, Polish cryptologists decrypted 410 signals: from Soviet General Mikhail Tukhachevsky, commander of the northern front; from Leon Trotsky,
and in fact:
The Russian staffs, according to Polish Colonel Mieczysław Ścieżyński, "had not the slightest hesitation about sending any and all messages of an operational nature by means of radiotelegraphy; there were periods during the war when, for purposes of operational communications and for purposes of command by higher staffs, no other means of communication whatever were used, messages being transmitted either entirely ("in clear," or plaintext) or encrypted by means of such an incredibly uncomplicated system that for our trained specialists reading the messages was child's play. The same held for the chitchat of personnel at radiotelegraphic stations, where discipline was disastrously lax."
So we can infer that the possibility that the Russians were using a cipher as devastatingly primitive as a Caesar shift is quite high - but that it is also just as likely that in reality, communications were intercepted in the clear. The cause seems to be likely down to poor operational security from the Russians. It seems also likely that the Soviets did possess comparatively sophisticated ciphers for the day; but that they didn't use them as a result of poor signals discipline.
It would also appear that an Ian Johnson has done some work on this exact subject in a paper titled "No Divine Intervention: A Reexamination of the Battle of Warsaw, 1920" which I was unfortunately unable to source at this time.
In conclusion not only does it seem likely this tale is accurate, it probably does not even represent the extent to which the Poles had compromised Russian signals. There seems to have been quite a bit of research done on this particular penetration of Russian communications networks during this time.