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 <A HREF="https://eudml.org/doc/109572">here</A>.