Grant's speculation opens up two distinct possibilities here, each of which opens a whole can of worms for the Federals.
1.
Grant's Fear/Lee jumps Butler - Grant has to cross both the Chickahominy and James, neither of which have readily available bridges and thus require all of his engineering assets for pontoon bridges, while marching over poor roads hemmed in by swampy terrain. In fact, the pontoon bridges the Army of the Potomac will build will hold the record for world's largest until 1945. Grant's supply trail, over fifty miles of wagons, will also have to bypass the crossings at Long and Jones bridges over the Chickahominy in favor of the longer route at Cole's Landing, this done to prevent an encumbering of the Army's infantry. In comparison, as Grant notes, Lee has a shorter route over better roads with all of his river crossings done by already built bridges; the Confederates hadn't needed to burn the bridges under their control, after all. Given all of this, Grant is exactly right that, if Lee discovers whats going on sooner, he (Lee) can move to the James and bash Butler good long before Grant can get their.
I have yet been able to find returns for the Army of the James at this period, beyond an uncited statement that Butler had 40,000 or so near Richmond by the end of June. At the start of the Bermuda Hundred campaign he did have about 33,000 so that will serve as the upper amount. Between what Beauregard has on hand and Lee moving the Army of Northern Virginia to link up with the Confederate defenders, you'd probably see between 60,000 and 70,000 Confederates; end of June returns show 55,000 in the area, and we know Beauregard took about 4,000 casualties during the mid-June timeframe. So Lee will outnumber Butler around 2:1 and has a few days before Grant can really come online to aid the Army of the James. I think it's fair to reckon Lee will smash him good.
2.
Grant gets jumped - Lee has the mobility advantage, and Grant is spread out over dozens of miles hemmed in by the aforementioned terrain, his own baggage and two rivers. Entirely possible and feasible for Lee to still send reinforcements to Beauregard while taking the majority or, if he is willing to risk it, the entirety of the Army of Northern Virginia with him and falling upon Grant's dispersed column. Lee could allow Grant to be in the middle of crossing one of these rivers and, when there are fewer on the Northern side of the river than Lee, they can get pounced on while Grant is unable to rescue them. For example, say Lee waits till Grant is crossing the James and has around 30,000 or so still waiting to cross, Lee could hit them and destroy a third of Grant's Army right there.
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Interestingly, E.P. Alexander's own memoirs contain a speculation on Lee's opportunity here in line with Grant's thinking, but of a less dramatic opening. Alexander, like Grant, realized that the infrastructure and terrain was in Lee's favor, thus granting him the advantage of mobility. Instead of envisioning an offensive to bash Butler, however, Alexander speculated that the Army of Northern Virginia could've went on the defense to inflict a sort of "Super Cold Harbor" on Butler and Grant as he came into play. This isn't exactly idle thinking either, as during the thick of fighting Beauregard did manage to inflict 10,000 casualties upon the Federals to 4,000 of his own at a ratio of 2.5 to 1, a rate almost exactly that of Cold Harbor. This despite Beauregard being massively outnumbered to the extent there was often five feet per man in the trenches, while most of his command was either militia composed of young boys and old men or recovering wounded. This is understandable,
as morale among the Federals was absolutely shot to the point there was serious risk of collapse of capability in their forces. Lt. Col. Theodore Lyman, Meade’s Aide-de-Camp, noted the route of 30,000 Federals by just 8,000 Confederates on June 22nd as a clear example of the exhaustion afflicting the Army, while the day previously Meade himself, in a letter to his wife, stated that Army of the Potomac was in desperate need of rest lest its morale utterly collapse.
All things considered, and given the need to protect Richmond, I'd imagine Lee would go with option one and hit Butler hard before retiring to entrenchments as the Army of the Potomac began to arrive on scene. Between his offensive strike followed by a much more stout defensive action than Beauregard could manage, I'm entirely certain an extra 30,000 Federal casualties could be inflicted. Using the 2.5 achieved by Beauregard and Lee at Cold Harbor as a baseline, that'd mean 12,000 Confederate losses. End of June returns show 96,000 Federals to 55,000 Confederates, so in this ATL it'd be 66,000 Federals to 43,000 Confederates. At this point, Lee can do quite a bit and has many options available to him.
Sheridan by this point has already got slammed by Hampton at the Battle of Trevilian Station and at Saint Mary's Church, while Early has done the same to Hunter at Lynchburg, ending the western threat to Richmond and effectively clearing the Valley by forcing Hunter into West Virginia. It's certainly a strategic opening for Lee to recall both Hampton and Early, adding around 20,000 men to his own forces around Petersburg and thus giving him near numerical parity with Grant. That, combined with the exhaustion and shattered morale of the Yankees gives Lee an opening to destroy the Army of the Potomac as a fighting force, at least for the remainder of 1864.
Instead of the above, he could keep Early moving into the Valley and threatening Washington. The defenses there are bare, stripped by Grant previously, and thus just as ripe for attack ATL as IOTL. Unlike historically, however, the Army of the Potomac isn't in an opening to detach forces to retrieve the situation around the city in the face of Early. This means that either Early will take Washington and burn it down before withdrawing back into the Valley, inflicting a grievous blow to overall Federal morale at a time it can't be afforded, or Grant will still have to give up some formations for the defense of the city. If he does that, presuming an equal number as to OTL of nearly 20,000 or so, than we're back to the above in that Lee and Grant's reduced hosts are near equal at 46,000 to 43,000. Lee can then plausibly detach 13,000 men, a force slightly larger than Longstreet had for Chickamauga in 1863, for duty in the West and
still maintain a better force ratio of 1.5 than the historical average of 1.65 around Petersburg.
What would 13,000 men mean for the West? Well, as previously stated, that could be very decisive for Peachtree Creek if we accept the proposition that nothing in theater changes there until the arrival of the Confederate reinforcements. Schofield was being held in place by a Cheatham's Corps, MacPherson was being held at bay by the Confederate cavalry under Wheeler and Thomas had two Confederate Corps coming down on him with the hope to knock his teeth out. An additional Corps at the right time opens up quite a few possibilities, such as it being used to pin MacPherson in place while Wheeler is used to to hit Thomas in the rear and/or, if the Confederates really want to push for it, cut the fords over the Chattahoochee; now the entirety of Sherman's command has just lost its ability to retreat and resupply.
Basically, lots of possibilities, all of them opening up for great Confederate success in both theaters.