Here's what John Milnor writes about this, in his collected works:
The projected Part II of this paper was never written. In fact I am chagrined to discover that I have never published any details about some of the announced results which were intended to appear in it. I was very grateful when Thom gave me permission to reprint his 1959 Bourbaki lecture, "Travaux de Milnor sur le cobordism", which gives a better account of this work than anything which I have published.
(page 249 of Part III of the Collected Papers of John Milnor, AMS 2007).
The paper by René Thom which Milnor mentions can be accessed here.