Hex editor (if you don't know what this is, then it's probably best you don't make these changes)
Text editor

Create a new save game, with the map you wish to use and the settings you like. Set the year to 1960. Save it with a name that does not include "1960".

Locate your save game (<steam folder>\SteamApps\common\SovietRepublic\media_soviet\save\<name>).
Open the file "header.bin" in your hex editor.
Search for the integer value 1960. In the save I have used, this has always been at offset 19C, but yours may vary.
You'll see the hex values A8 07. Change this to 9E 07 for 1950. Save the file.

Open the file stats.ini in your text editor.
Search for the string "1959" and replace with "1949". If you made the save as soon as the game began then there should be two of these entries. It is possible that by random chance "1959" could appear in other places. Only edit lines starting with "$DATE_YEAR".
Update: Now that seasons have been implemented, the game no longer starts at the beginning of the year, so the first year may be "1960". If you don't find "1959" then try "1960" and replace that with "1950".


Save the file and then open the save in the game.

You can use other values too. Even though vehicle availability dates start at 1950, they still appear even in 1940.
Update: Vehicle dates are now strict in the open beta, so you will need to use mods or edit the start dates of the vanilla vehicles.

The modding community has created this collection of pre-1960 vehicles to use:
https://psteamcommunity.yuanyoumao.com/sharedfiles/filedetails/?id=2354857994