While no one translated Soviet newspapers into English in full, you do have some choices for these kinds of primary sources in English.
А weekly digest of English translations of selected major stories, primarily from official dailies Pravda and Izvestiia, but occasionally from other sources as well, including other major newspapers (e.g. Komsomol'skaia pravda, Literaturnaia gazeta, Sovetskaia Rossiia, Krasnaia zvezda, in the 1980s Argumenty i Fakty), trade publications (e.g. Sotsialisticheskaia industriia, Uchitel'skaia gazeta), and even some regional newspapers. Not all stories were translated in full: some were condensed or excerpted. Each issue also included a thematic index to Pravda and Izvestiia for the week. Yes, the title seems ridiculous now.
Unabridged translations of selected items from national newspapers, focusing on Pravda and Izvestiia. Often complements Current Digest of the Soviet Press, either by providing translation of items omitted from Current Digest, publishing a full translation where the Current Digest only included an excerpt, of offering a different translation. Because this publication came out only bi-weekly and because it included unabridged translations, the number of items covered is far smaller than in Current Digest.
Includes English translations of selected stories from a variety of Soviet newspapers, from major dailies and weeklies to trade and local publications, but resource is organized around topic, not publication source, which means you can only browse a particular publication if you already know the title. Not all stories were translated in full: some were condensed or excerpted. Clunky interface.
Moscow News was Soviet Union's English-language weekly aimed at readers abroad and ex-pats in the Soviet Union, published since 1930.
Daily newspaper published by the Press Department of the Soviet Embassy in London.
An English translation of Pravda was issued for issues between 1985 and May 5, 1988.