This, I think, would be an even bigger development then a temporary lifting of the blockade around Hampton Roads. Johnston is without a doubt at least an equal of McClellan, has a Confederate Army that hasn't been bloodied by two different Northern invasions, and the natural terrain of Virginia means he has plenty of river lines to use on the defense; in short, Johnston will be in position to make this Alt-Overland Campaign far too bloody for the North. Even presuming he isn't able to replicate and successfully execute something like he tried to do at Peachtree Creek IOTL, he'll subject the Army of the Potomac to such attrition that an Anglo-French intervention is probably likely, or the Democrats winning the Congressional elections that fall at least.
It doesn't help that McClellan will no doubt be as sluggish here as he was on the Peninsula IOTL, never mind the fact that he would carry out this campaign very reluctantly. Unlike Grant IOTL he will NOT fight it out on this line if it takes all summer.With McClellan's reputation? Poorly.
My imagination is that he advances on the Rappahannock, and Johnston manages to mousetrap him somewhere, which leads McClellan to retreat, and Johnson is simply holding the line. McClellan might then try an overland through the Valley aimed at Culpeper Court House, but would again probably be foiled there too. I'd see McClellan being defeated at every crossing, and giving way to a solid stalemate in 1862, with McClellan perhaps sacked earlier than OTL.
This is where the POD really generates a LOT of Butterflies. The aftermath of a post alt-Overland campaign has the potential to go in a lot of directions. Lee may not get the chance to command the ANV for example.