Depends on the scale of the Union defeat, are you imagining a withdrawal from Pittsburg Landing after Day 1 or are you imagining a total destruction of Grant's army? For the former, the Confederates have won a signal victory. They inflicted heavier causalities and possess the battlefield. On the Union side, the defeat is embarrassing and there will be calls for Grant and Sherman's heads, probably louder than OTL. Interestingly, IOTL, Halleck was given a free hand by Stanton to fire Grant and Halleck, despite his behavior, surprisingly opted to cover for Grant and put the blame on regimental commanders. I'm not sure if Halleck would still cover for Grant or not, but Grant's long-run career is definitely put in a more precarious position. It's possible that Halleck would "promote" Grant into that useless second-in-command position.
However, from Pittsburg Landing itself, the Army of Mississippi can't really exploit their victory. They don't the transport vessels to carry them across the Tennessee River to engage Buell's Army at Savannah. I find it likely that Halleck would be reorganizing the Union armies at Savannah, safe from any potential Confederate attack. That said, I'm not sure if the Confederates would continue an offensive strategy or if they would go on the defensive. After all, Halleck is still poised to cross the Tennessee River with a bigger army and is close to Corinth.
For the latter, a complete destruction of Grant's Army of the Tennessee presents a stunning reversal of fortunes in the Western Theater. Grant and Sherman are gone while Halleck and Buell have control of things. The destruction of the Army of the Tennessee means that the Confederates have achieved numerical parity in the western theater. The door is open for the Confederates to launch a counteroffensive to retake Tennessee and possibly re-establish a foothold in Kentucky. However, I think that New Orleans will still be lost. I doubt that the troops sent from New Orleans would be returned when such a stunning victory has been won and the opportunity to reclaim lost territory has come.
Then again, I note that complete destruction is... kinda out of the picture. Aside from the excellent defensive terrain around Dill Branch and Tilghman Branch and the reinforcements from Buell, the Army of the Tennessee always had the option of withdrawing across the Owl Creek and regrouping at Crump's Landing. Still, if Pittsburg Landing could be overrun, the Confederates could bag a lot of dispirited Union troops taking up shelter along the Tennessee river.