The Schwartz Kapelle would launch its coup very soon afterwards. They don't have to do it "on the fly": Goering will succeed Hitler, and he is a highly discredited and vulnerable figure. On 20 July they had to act immediately because they were exposed for the assassination, and Goering and Himmler were at large. At least that's AIUI. Stauffenberg got the bomb into Hitler's presence on 11 July and 15 July, but aborted both times because Himmler and Goering weren't there. By 20 July, they decided to act regardless, and hope they could overcome Goering and Himmler.
(Actually, there's a perfectly good WI: Himmler and Goering are present on 11 July or 15 July and the bomb gets all three targets. Then VALKYRIE almost certainly succeeds.)
On 23 June... the Allies were recovering from the Great Storm. US VII Corps was closing in on Cherbourg, while the rest of US 1st Army was slogging through the hedgerows, and British 2nd Army was stalled at Caen.
But anyway. Goering settles in, will immediately try to open negotiations with the Allies - and get nowhere. The SK will schedule their coup for no later than 7 July. (There will be changes as additional Heer commanders sign on explicitly.) On say 7 July, the SK moves. They need to arrest Goering and Himmler. Goering will be in Berlin. His personal bodyguard will be Luftwaffe field troops, so it will require a significant military effort to take him. Dunno where Himmler would be... He wasn't in Berlin on OTL 20 July. Maybe he will be in Berlin working on his own anti-Goering coup.
Or if the SK are really lucky, Goering would have him arrested. (Just how bad was the feud between them?)
Would Goering perhaps hide out at Karinhall instead of Berlin? Or at a Luftwaffe base with a lot of troops around him? (Possibly for fear of Himmler?)
If Himmler starts plotting immediately, and approaches figures in the Heer, the SK will know of it. In which case they can either turn him to Goering, or let him do the dirty work. I don't think there's a significant risk of Himmler actually recruiting any Heer leaders.
In an ideal scenario, Heer leaders who are actually with the SK pretend to back Himmler. Relying on this purported support, Himmler attempts a coup against Goering by sending SS forces to get him. There is a large pitched battle in which Goering is "accidentally" killed, following which the Heer moves to arrest the traitor Himmler, who is "accidentally" killed in the gunfight with his SS bodyguards.
Note however that the SK would have to wait for such a scenario to develop.
As to strategy - IMO the SK or Goering will initiate several strategic withdrawals. Possibly from Italy; fall back to the Alps and let Mussolini fend for himself. Western France, starting in early July, most likely. Although the Germans didn't anticipate the sweeping breakthrough of COBRA... But absent Hitler's rigidity, they will surely evacuate the west in late July.
Finland moved to get out in August; probably sooner.
German withdrawal from Estonia and Latvia. German withdrawal from Norway? (What's the point of occupying Trondheim with the Allies approaching Warsaw and the Rhine?)
U-boat operations halted? (The Germans are losing a U-boat for every freighter sunk.)
V-weapons stopped? Wasteful, but Hitler craved the prestige.
Greece evacuated in August too.
Romania will sense it's being left on its own, and flip sooner.
Hungary's in an odd position.