How long did Bayonet Melee engagements last during the napoleonic wars, and how many casualties would there be?

Usually, zero minutes, zero seconds.

Man-to-man combat is so terrible that without defensive arms, it is almost impossible on the battlefield. When one battalion charged the other with fixed bayonets, the outcome depended on the effectiveness of the defenders' fire. If successfully held until clothing-burning range, the impact could be so devastating as to stop the charge in its tracks, leaving them vulnerable to a counterattack. Both in the case of a successful charge and a counter-charge, though, bayonets rarely crossed, and it was quite rare for anyone to even get stabbed. What usually happened was that the troops under attack took to their heels when the enemy got too close, fleeing in greater or lesser disorder. In terms of casualties, there would usually be few, most of them prisoners; men almost always throw down their arms and raise their hands in surrender before getting stabbed.

Bayonet fighting was more common when a position -earthworks, defended villages, woods- were being taken by storm, or when weather rendered fire impossible, but it was still typical that most would flee before engaging in a melee. While house-to-house and room-to-room fighting was present in most Napoleonic battles, they didn't produce that many stab wounds: data is pretty sparse, but all I've seen indicates only a single-digit percentage would suffer actual stab wounds.

I mentioned weather putting muskets out of action earlier; as it happens we do have an account of such fighting that does actually give a time for the melee of one French battalion against two Prussian. At the Battle of the Katzbach, the rain was so bad that the Prussian 8th Brigade could barely see the enemy as they advanced; the battalion on the right wing stumbled into a French battalion to their right, which formed square.

We now doubled our pace, lowered our muskets, and attacked the middle square of French grenadiers with fixed bayonets amid terrifying cries of “Hurrah!” The square stood like a wall. We came within two paces. For a moment our people stood across from the French so that both sides could see each other. We officers shouted: “On them, on them!” and now the soldiers reversed their muskets and drove into the French with the butt of their muskets. Because we stood in line, the square was quickly encircled on the left and right and thus attacked from all sides with the bayonet and the musket-butt. No one thought of giving quarter; after ten minutes the entire square was shattered and transformed into a pyramid. Some 150 unscathed and lightly wounded found themselves among heaps of dead and wounded; they were sent back as prisoners.

Complete casualties for the French battalion are unknown, but the Prussians lost 190 men killed or wounded and captured 170; it's unlikely the French had any fewer dead in the melee, so they probably suffered 400 casualties or more all told.

Clearly, this is an exceptional circumstance, but it is a good illustration of how savage close combat could be.