There are a few ways to do this. Ordinarily you would want the mica joint to be as short or in a visibly appealing location as possible. I'm just mentioning this for you to keep in the back of your mind while planning what you'll do.
Since you didn't state which way the 90 or 60 was or the depth of the three directions, I'll just give an overview of how I'd do it. First take the substrate and cut the depths of the three lengths. Fit them and scribe them to fit the walls.
Then take them out and lay them flat. You could make connections at the places where you want substrate joints underneath with FASTENERS.
Or, make a scabbed cleat from a substrate to screw in from the bottom to hold the joints together. Actually, it is better to do both because the added size of the cleat will provide more support.
In doing these substrate joints, keep in mind you don't want the mica seam over a substrate seam. So, once the top is scribed, you can laminate a section having the seam of that piece fall short, so the next piece will fall on solid material. IOW, if you want the two mica corners to be mitered, layout the substrate to be joined at a 90 degree at the point of where the "U" forms. Or, the other way around, where you have the substrate made to join on a miter (which would be a longer joint, and less structural), layout the mica to be at 90 degrees.
If you don't prelaminate any pieces, fit and mark the mica where they join with pencil marks across the seam so when you laminate them after the substrate is installed, they will match up. A word of advice. If, for example you will be laminating three pieces, do it in an order, like from left to right, or right to left. Your joints will work and fit better. I wouldn't laminate before installing the tops.
A trick about seaming and fitting mica joints. Mating edges can be straight edged by using a straightedge and a router with a trim bit with a bearing. Or, two pieces can be seamed that will be joined by placing them together, and running the router between both at the same time, so any differential will happen to both allowing them to fit perfectly. For that, you'll need to fabricate a jig to clamp both pieces and have a straightedge to guide the router.
Or, scribe and install the entire top with the substrate joints where you pre planned them to be. Then starting on one end (either right or left) Scribe and fit the mica and layout the seams at the appropriate locations. In doing it this way, have all the mica seamed and scribed ready to go, and glue up (with contact cement, and I prefer solvent based), the substrate and the mica at the same time. Then when it has flashed off, clean the seamed edges (you can rub your finger across the edge to clear any glue), lay down dowel rods as spacers (about 10" to 12" apart), and when the sheet(s) are in position do one at a time. A little trick about getting tight joints in mica: When seaming one sheet to another, assuming one is stuck down, touch the edges together and leave one dowel rod under maybe about a foot away, and pull out several after that dowel but don't press the mica down hard, just slight pressure, and there is a slight allowance of space. That will give a tiny bit of force towards the joint as the sheet is pressed down.
This is kinda tricky because if the dowel is too close to the joint or too much is allowed, you may not be able to press it down. That particular dowel can be a 3/8" so it won't raise up the gap too high. For the other dowels you can use 3/8" or 1/2" dowels. I prefer 1/2".
Countertop mica is usually 1/16" thick and is much easier to handle than vertical grade which is 1/32".