Star Trek (2009) and Star Trek Into Darkness were set before the events of the original series on either timeline.
On the Prime Timeline, TOS lasted for three seasons which I think equates to three years.
Star Trek Beyond states that they are three years into their mission.
I'm sure that the mission would be vastly different given how events have diverged in the timelines. But in terms of absolute timelines (and personal/ interpersonal development) is Beyond set after the time frame of TOS? I am not sure about how different the start dates of the five year mission where in either time line.
Also is this deliberate? Kirk makes a remark about their lives becoming 'episodic' which makes me think that it is. But it would nice to have this confirmed somewhere.