I thought they picked them up after hearing about the Botany Bay from Spock Prime (though I could have that part wrong)? If so, they would have had a year at most...
No, I think Section 31 did its own research into the whereabouts of the Botany Bay
. I don't recall Spock Prime being identified as the source of the leak about the existence of Khan.
In this timeline, Harrison could have been around in the 23rd century almost as long as Kirk!