At first glance (although difficult to see with the upside down photo) my first thought was fireblight, but Mulberries don't get fireblight.
Knowing this I suspected some form of bacterial blight. A quick search turned up exactly that...bacterial blight. Bacterial blight comes in several different flavors. The closest photos resembling yours is Bacterial leaf scorch. Leaf scorch behaves just as you say...browning from the tips and leaf margins moving back until the entire leaf turns black/brown.
I thought that was probably it until I read your post about the wiggly roots and it's feeling not well anchored. Even worse you said it leaned on it's own.
I'm thinking almost for sure there is some root issue at play here. Root rot is very common in Mulberry trees, but with your weather being dry it seems less likely (though not impossible). Have you watered this tree? If so, how often and how much? Does the tree reside in low lying ground where it may be subject to wet feet?
Have you recently applied a herbicide to your lawn like glyphosphate (roundup), 2,4D, or any broadleaf weed killers? Has your neighbor? It doesn't look like that kind of damage at all, but you never know.
Do you have issues with moles/voles? Plants that suffer from too little water uptake (this does not necessarily mean too little watering) often begin dying from the top down because as the leaves transpirate the lack of water is unable to reach the top and tips adequately enough to maintain life. Too little water can be caused by root rot or some other root disturbance caused by burrowing animals etc. Ironically too much water (wet feet) will often exhibit symptoms of too little water in almost all plants I know of. If adequate moisture is available and the roots are damaged by rot or whatever, the roots cannot deliver water to the tree.
If I had to guess, I'd think it's a bacterial infection of the roots which has also manifested itself within the tiny trees canopy. Either way or anyway it's not good. Not good at all.

Prune away the damaged area well below the damaged tissue and discard (do not compost) it to the trash. Stick your finger down into the soil (if you can) and make sure there is moisture there...but not wet.
Be prepared to lose that tree...you may get lucky and it may recover, but looking at the photo on a young tree like that I can tell you it doesn't look very promising.

Stay tuned for other responses. There are folks here who grow Mulberry and are familiar with it (I am not).

Thank you for the response. Sorry the picture came through upside down. It looks ok on my end but I'm posting from an iPad and the photos it takes often have problems with orientation.

I googled bacterial leaf blight and you're right, the symptoms do seem to match. The USDA says mulberry is a known host and the disease clogs up a tree's xylem, restricting water flow. It makes sense then that the symptoms of the disease are similar to the symptoms for water uptake problems. I really hope it is "just" root problems and not that disease. It sounds terrifying! O.o I did not know that mulberries get problems with root rot so maybe I brought this on by accident.

Here is another pic showing the bed this is in and the other mulberry. (Hopefully this one comes through right side up....) My native soil is a dry riverbed. It has excellent drainage but is too rocky to dig so I built the raised beds on top of it with cinder blocks. This being the trees' second year and with them growing so well without interference, I was not watering them very often for most of the summer but kept an eye on them. I really only started watering this one more deeply when I staked it up, because I thought it would help the roots recover if I gave it some extra care. Since then and depending on how hot it is each day I've been watering it more. Every other day when it was in the 90's. Every 3-4 days when in the 70-80's. I try to go by how dry the soil is. I don't measure, though I'd say a few gallons each time. Enough to penetrate the soil without staying soggy. I have been watering the other one at the same time but not as many gallons.

I haven't applied any chemicals. The neighbor uses herbicide on their driveway but as far as I know not on their field. We have well water so there is no chlorine or anything. We've had two or three mole holes pop up but not near the garden and there has been no new activity when I stomp them down so it doesn't appear that they've colonized enough to be an issue.

The more I think about it the more I suspect that the combination of root damage and a sudden increase in watering are to blame. I would be very glad to hear what others have to say as well.

Here is a link that might be useful: Bacterial leaf scorch reference

Cut back on your watering a bit and see what happens. Mushroom compost is like a sponge, and on top of that hard riverbed it may be laying too wet. Don't forget to cut away that damaged tissue...it's dead and it's not going to recover no matter what the twigs look like. Get rid of it.

Sounds good. Yesterday and today the dieoff has slowed considerably. It has been four days since the last watering. I am crossing my fingers that it could possibly be this simple.