Reviewed the timing of the battle a bit (all per Shattered Sword):
0715: Order given to begin re-arming procedures. Given perfect ops, would be complete by roughly 0845, with 45 minutes to spot and launch.
0728: Tone reports to Tone--10 surface units
0745: Nagumo recieves Tone #4's 0728 report
0747: Akagi orders Tone #4 to stay and amplify contact
0800: Estimated time for Tone #4 to spot American carriers if launched on time and flown path as ordered. If launched late and flies path as ordered, first sighting will be even later.
0800: CAP composition is down to 9 planes, four of which (Kaga planes) are setting up to land. Shortly to be bolstered by additional planes (about 7) but CAP is being kept small given expectation of ongoing landing and spotting ops soon.
0815: Tomonaga's strike begins orbiting Kido Butai.
0820: Tone #4 finally spots and reports American carrier-first solid confirmation of American carrier's present.
0837-0912: Tomonaga's strike is recovered.
Playing out a similar timeline with all of Tone's reports shifted 30 minutes later:
0715: Nagumo orders second group armed for ground attack. Will be completed by 0845, ready for spotting once Tomonaga finishes landing.
0800: Henderson's squadron attacks for little effect, but disturbs CAp (down to 9 planes, with four landing). Tone #4 transmits spotting 10 American surface ships.
0815: Tomonaga arrives over Kido Butai. Nagumo receives Tone #4's 0800 spotting report, but believes (as he did initially in OTL) that it is primarily a surface force even if unseen carrier present with it. Decides to continue arming for land attack for strike on Midway after Tomonaga lands.
0837: Tomonaga's force begins landing.
0845: Second wave completely re-armed for ground attack.
0850: Tone #4 reports presence of American carrier. Position reports are spotty and inconsistent. Nagumo is faced with challenge--deck is busy for at least another half hour (particularly given ongoing trickle of American planes). Rearming would push his earliest strike time out about two hours (1050) at best, probably past 1100, while if he begins spotting once Tomonaga lands, he can strike with all four carriers on either Midway or the American carrier, which may not even be where Tone #4 is reporting.
The questions for him: Spot at 0915 for a ~1000 strike with land-attack weapons, and if so on what target? On Midway as planned, or on the single (possible, probable, position poorly known) carrier? Or rearm, pushing strike to past 1100, but giving time for additional scouts to amplify and sort out Tone #4's spotty reporting?
It's interesting: if he re-arms, then the situation essentially just converges with OTL until after the 1020 strikes: no planes on deck except CAP fighters, hangars full of fuel and bombs, etc. However, if he spots immediately, he'll be launching right around 1010, but his CAP will have been depleted by almost an hour of American attacks. It's possible that, like at 0800, his CAP will have drawn down in preparation for spotting starting at 0915, possibly down to a similar ~9-15 planes, which may have shot their cannon ammo dry in the meantime at Waldron (0920) and VT-6 (0940). When VT-3 arrives (1000), the CAP will be scattered and low on fuel and ammo, and may be much more ineffective in responding.
It's unlikely to allow any hits given the poor quality of the planes and torpedoes, but it likely prevents the kKido Butai from launching their ground-attack-armed strike as they maneuver to avoid attacks and cannot come into the wind. Thus, we actually find the situation poular legend would say exists: all four Japanese carriers bare minutes from a strike, either on Midway or on the Yorktown at 1020 when SBDs begin dropping through the clouds en masse.