Kentucky and Kansas have chased each other for the distinction of tallying the most all-time victories for decades. Their ascent to the top spot coincides with winning their 1000th victory.
Kentucky claimed they took over the #1 spot as the winningest team in college basketball on January 6th, 1969, beating Mississippi State. It was a three-way race to 1000 wins, but with Kansas and Oregon State, not North Carolina. The milestone was disputed when Kentucky announced it had reached 1000 wins because five wins came in the World University Games in Tel Aviv, Israel. The Wildcats won the five game tournament, and Coach Adolph Rupp counted them among the all-time victory total. Oregon State responded with a claim they had wrapped up the 1000 victory total with 27 wins during an International Tournament of Australia. The NCAA ruled that international competition for both teams was not considered part of their victory total, despite the fact Rupp continued to count them.
Meanwhile, Kentucky claimed they searched their archives and found records of victories over the Lexington YMCA, Kentucky University (not a typo) and Georgetown College. Again, they claimed more victories than Kansas and announced they reached the 1000 win plateau, much to the frustration of Kansas and their head coach, Ted Owens.
Oregon State suffered a down year and was out of the race. At the same time, Kansas was losing games and their two game lead over Kentucky in total victories was in jeopardy. KU lost to Missouri and Iowa State, while Kentucky marched on.
Reliable information suggests that Kentucky gained the number one spot no later than January 27, 1969 against Georgia. They reached the milestone ahead of Kansas, which is the only indication of exactly when they surpassed the Jayhawks for the top spot.
And so, a quick answer to your question indicates that January 1969 was the approximate date, and the Wildcats never looked back!
Please note: While this is the date I know Kentucky to have assumed the top spot, it is not impossible for them to have occupied the position earlier and lost it to Kansas along the way. I am aware they took over for good in 1969, but I am less certain about anything much before then.
Thanks for asking!