I have to admit that I simply cut and pasted from Wikipedia’s article on the USSR State Motto, so you can find more there. The quick guide is as follows:

Пролетарии всех стран, соединяйтесь! – Russian, of course.
Пролетарі всіх країн, єднайтеся! – Ukrainian – the letter ї is a giveaway.
Пралетарыі ўсіх краін, яднайцеся! – Belarusian – the letter ў is the clue here
Бутун дунё пролетарлари, бирлашингиз! – Uzbek. I wouldn’t have got this; most of the letters are fairly standard except for ё (which you do get in Russian, but this is obviously not a Slavic language).
Барлық елдердің пролетарлары, бірігіңдер! Kazakh. The letters ң and қ leap out here, but both are used in other Cyrillic Turkic scripts.
პროლეტარ ყველა ქვეყნისა, შეერთდით! – Georgian, obviously.
Бүтүн өлкәләрин пролетарлары, бирләшин! – this of course is now obsolete; Azeri is now written in the Latin rather than Cyrillic alphabet. I guess this would now be written “Bütün ölkələrin proletarları, birləşin!”
Visų šalių proletarai, vienykitės! – Lithuanian, as well knows. The ų and ė are characteristic.
Пролетарь дин тоате цэриле, униць-вэ! – causes some confusion for people who go through the letters and realise that it is actually Romanian in Cyrillic, or Moldavian as it used to be called. I was puzzling a bit over цэриле, and then realised that in Romanian it is probably ţerile which makes both the Cyrillic transcription and the derivation from Latin terræ more obvious.
Visu zemju proletārieši, savienojieties! – Latvian. The letter ā is a crucial one here.
Бардык өлкөлордүн пролетарлары, бириккиле! – Kyrgyz, though I have to admit I would only have been able to work out that it is one of the five Turkic languages. Wikipedia claims that ү is now only used in Kazakh but obviously that isn’t the case.
Пролетарҳои ҳамаи мамлакатҳо, як шавед! – Tajik, the only Persian language on the list. Also the only one with the letter ҳ.
Պրոլետարներ բոլոր երկրների, միացե’ք! – Armenian, obviously.
Әхли юртларың пролетарлары, бирлешиң! Kyrgyz. Note use of ң (like Kazakh) and ә (like old Azeri).
Kõigi maade proletaarlased, ühinege! – Estonian. õ is the crucial identifier.

That’s the fifteen official languages of the USSR’s republics, all now independent states. The bonus two were languages spoken in autonomous parts of the Russian Federation:

Kaikkien maiden proletaarit, liittykää yhteen! – Finnish, as spoken in Karelia
Барлык илләрнең пролетарийлары, берләшегез! – yet another Turkic language, in this case Tatar. Barlıq illärneñ proletariları, berläşegez!