woldsmandriffield: The position after 72 Ke2 is analysed by Levenfish and Smyslov in their book on Rook Engings (pp.112-3). The difficulty of the ending is shown by some misevaluations by the authors.
Before we get there though, Roselli could have drawn easily with 72 Rg6. What can Nimzovitch do? If 72..e4 then 73 Rg4 and if 72..Ke4 73 Rg4+ Kd3 74 Rxh4 e4 75 Rh8 is a draw.
Instead White put his King right in front of the advancing e-pawn and played 72 Ke2 which L&S considered to be losing. Nimzovitch answered with a check 72..Ra2+ And now since 73 Kf3 runs into e4+ and 73 Kd3 or 73 Ke3? lose the h-pawn, Rosselli had to decide where on the first rank to retreat with his King.
He chose 73 Kd1 which loses. Before analysing the alternative options which both draw (Kf1 or Ke1), it should be noted that White can in fact also draw with 73 Kf3. L&S give the line 73 Kf3 e4+ 74 Kg4 Rg2+ 75 Kf4 Rf2+ as a win for Black but but White is OK after 76 Ke3 Ra2 77 Kxh4. Simple as that.
Whiteshark has posted a diagram of the position that arose after 73..Ke4 74 Rxh4+ it is a loss with the White King on d1 but a draw if the King stands on either e1 or f1. The reason is that after 74..Kd3 Black threatens mate and gains a tempo and after 77 Kf2 e3+ in the game he gains another.
With the King on e1, after 74 Rxh4+ Ke3 74 Kf1 there will be no check after a pawn advance and likewise, with the King on f1.
Nonetheless, L&S thought that 73 Kf1 was still a loss. They give 73..Ke4 74 Rxh4+ Kf3 75 Kg1 e4 76 Rh8 e3 77 Rf8+ Ke2 78 h4? Ra4(!) 79 Rh8 Rg4+ 80 Kh2 Kd3 81 Rd8+ Rd4
But instead of the premature advance of the h-pawn, White can defend with 78 Re8! and now two lines:
A) 78..Kd2 79 h4! (now is the right time for Harry to go forward) 79..Ra1+ 80 Kg2 Ra2 (80..e2 81 Kf3=) 81 Kf1
B) 78..Ra1+ 79 Kg2 Ra3 80 Rb8 Ra2 81 h4 Kd1+ 82 Kf3 e2 83 h5(!)=