The Fallout 4 mod sharing feature for Xbox One will be available in early 2016, according to Bethesda Game Studios' Twitter account. The studio also revealed that they hope to bring the feature to PlayStation 4.
During Microsoft's E3 press conference, Todd Howard, Game Director and Executive Producer at Bethesda Game Studios, revealed that mod sharing would be possible for PC and Xbox One.
#Fallout4 mod sharing (from PC) coming after launch to Xbox One in early 2016. We hope to bring it to PS4, too. #BE3— BethesdaGameStudios (@BethesdaStudios) June 15, 2015