In terms of the events of July 20, it shouldn't actually make that much difference. You'll have the plotters insisting Hitler's dead, as they did OTL, with Keitel and Bormann insisting he's still alive to all the military commanders and Party officials they contacted in the hours after the explosion, just like OTL. They didn't actually provide proof to the nation at large that he had survived until his radio speech, which iirc was sometime after midnight (it took that long for the broadcasting truck to reach Rastenburg from wherever it was - Konigsberg, I think?) - by the time that happened in OTL, the coup in Berlin had already been suppressed.
I suppose the big question is what Remer does - without that famous OTL telephone conversation with Hitler (I'm assuming he's too badly hurt even to croak something down the phone), can Goebbels win him over sufficiently for him to disobey his orders? Given Remer's Nazi loyalties and the accounts that suggest he was deeply suspicious of his orders even before the encounter with Goebbels, I guess he could end up acting as per OTL anyway.
Of course, the big difference is going to be afterwards, assuming Hitler remains incapacitated for weeks or months, it could give the commanders in both East and West more leeway to try different things to restore the situation. I'm not confident that there's much they can do to stave off defeat for very long at all, but the particulars of how the rest of the war plays out might be very different. Especially if the Reich ends up collapsing anyway as the various surviving hierarchs try and fill the vacuum Hitler leaves, with much in-fighting etc.