The sides have been cut out and I marked the center line and 1.1385" on each side of the CL.  

I used a try-square and drew the diagonals on the marks (at 45°).  

I drilled the roller hole and now I'm using a 45° table saw sled to cut the diagonals.  

TP Holder ends.   Please click for info on how I drill clean holes.   Also, you might want to see why the hole is 5/8".

This is 3/16" birch plywood, 5-7/8" long.   Top, bottom (1.7635"), and back pieces (2.277")cut to width (Len = 5.875"), ready to bevel.  

I mounted a 22.5° chamfer bit (half of 45° in the router table.   I pushed a piece of the material I want to bevel, as a spacer, up against the bit.

A little different angle, you can see the plywood is not up against the fence.  

I lay a flat ruler on top of the piece to be beveled and against the fence then adjust the router bit's height until it is just below (mises) the ruler.  

Here you can see the bevel bit will take off the lower corner and leave the top edge untouched.  
Note: I like Rockler bits, but they don't label them with size/angles only the Rockler number (which is most incovenient), so I mark them with a permanent marker so I don't have to look them up every time I need to use one.   They also change their numbers ocassionally, and consolidate lots of same style bits to one page (which hides the old number), so: I had to make a chart, of the bits I own, with their numbers and sizes/angles.

Bevels cut on tops, bottoms and backs.  

Gluing top to ends.   Note the 5-1/2" by 5-1/2" block used as an alignment frame.

Top being glued to an end, notice the bevel is toward where the back piece goes.  

All 3 back pieces being glued with a strap for tension.  

Direct from Walmart, a plastic toilet paper roller.   The shoulder on this one is just under 5/8" thats what determined the size of the hole in the end pieces, it can be changed for a different roller.   I can replace this with a nicer one when I find it.

Heres what it looks like installed.  

Sticks out a long way from the vanity.   Note the fake drawer front on the front of the vanity, behind that, and right above the current holder, is where I plan to insert the new holder.

Sticks into the storage area about 4", hangs on anything you try to store, especially extra toiletpaper rolls in plastic.