In Fallout Shelter it says, that radio studio increases dwellers happiness level. Does upgrade of this room increases this effect?
The radio room serves two purposes:
As @camelCase stated already, high Charisma increases the happiness of dwellers in your vault, as part of #1 above. Upgrading the radio room serves to point #2, in that it increases the chance of attracting a dweller. Expanding (adding additional rooms) of the radio room and putting additional dwellers into the radio room serves to make the cooldown for the search for dwellers to take less time.
So if you're interested in only keeping existing dwellers happy, high Charisma is what you want. (Plus probably a 3-room expanded radio room, so you can have the maximum number of high Charisma dwellers broadcasting)
If you're interested in finding dwellers in the wasteland, you'd want to focus on upgrading and/or expanding the radio room (or both) faster, and in my experiences, Charisma doesn't seem to have any affect on dweller-finding.
Based on my observations only, but it may answer the question somewhat:
Happiness appears to be related to the charisma of people in the radio studio. I have noticed drops when I take high charisma people out of the studio.
Having more radio rooms means more dwellers can be assigned, and potentially higher happiness (based on total charisma).
According to this Fallout Wikia page, the only bonus for upgrading radio studios is: