If you can scan your sketch, then you can import the digital photo of it into Vcarve. Thanks to you drawing it on graph paper it would be easy to scale to exactly the size you want. There is a trace vector option that might be able to do a quick job of turning it into vectors. if not, you can quickly draw over your sketch to create the needed vectors.
Each of the 5 unique pieces can be copied/edited from the initial master, then saved to their own layer so you can copy/edit the next one easily. Etc..
Once you have vector outlines of each piece, including the inside outline where the coins will end up, you can copy the duplicate ones, then lay them out on one sheet for efficient use of the material. It is easy to create toolpaths to profile cut through, around the outside and inside of each shape. Any shape that has an area/pocket you want to cut into but not through is just as quick using the pocket tool path. You can even quickly add 1/4" holes to each layer to help aligning them (with a short 1/4" dowel) when gluing them up. The holes can go through the inner layers, and half way through the two outer layers.
If
I'm guessing, but as I'm very familiar with the drawing tools in Vcarve/Aspire it might take me 1/2 hour total to create all the needed vectors if I had your initial sketch.
One piece, depending on which machine used, might take 2 minutes to cut around. A little more or maybe a lot less with an industrial machine. 2 minutes each for 11 pieces is 22 minutes. Shapes will be perfect. You might need to sand/trim off tabs that are used to hold the parts in place while being cut.
While the parts are still on the CNC bed, you could additionally round over the outer edges of the outer pieces.
Glue together, using short dowel pieces to help align all parts. Sand and finish.
While you are trimming, gluing, sanding, finishing the first one a second can be being cut on the CNC using the same file.
There will be time you need to add for clamping down the plywood, booting up the PC that runs the CNC, homing the CNC, then zeroing out X, Y, and Z for the project being cut. Add some time at the end for cleaning up the CNC and surrounding area.
Add some more time to upload/share images of and VCarve files for the design on your favorite web forums. 
If I was going to make just one, I doubt I'd use the CNC to cut the parts. If I was going to make two or more I definitely would take the upfront time to draw it up, then cut them all out using the CNC.
4D
Each of the 5 unique pieces can be copied/edited from the initial master, then saved to their own layer so you can copy/edit the next one easily. Etc..
Once you have vector outlines of each piece, including the inside outline where the coins will end up, you can copy the duplicate ones, then lay them out on one sheet for efficient use of the material. It is easy to create toolpaths to profile cut through, around the outside and inside of each shape. Any shape that has an area/pocket you want to cut into but not through is just as quick using the pocket tool path. You can even quickly add 1/4" holes to each layer to help aligning them (with a short 1/4" dowel) when gluing them up. The holes can go through the inner layers, and half way through the two outer layers.
If
I'm guessing, but as I'm very familiar with the drawing tools in Vcarve/Aspire it might take me 1/2 hour total to create all the needed vectors if I had your initial sketch.
One piece, depending on which machine used, might take 2 minutes to cut around. A little more or maybe a lot less with an industrial machine. 2 minutes each for 11 pieces is 22 minutes. Shapes will be perfect. You might need to sand/trim off tabs that are used to hold the parts in place while being cut.
While the parts are still on the CNC bed, you could additionally round over the outer edges of the outer pieces.
Glue together, using short dowel pieces to help align all parts. Sand and finish.
While you are trimming, gluing, sanding, finishing the first one a second can be being cut on the CNC using the same file.
There will be time you need to add for clamping down the plywood, booting up the PC that runs the CNC, homing the CNC, then zeroing out X, Y, and Z for the project being cut. Add some time at the end for cleaning up the CNC and surrounding area.
If I was going to make just one, I doubt I'd use the CNC to cut the parts. If I was going to make two or more I definitely would take the upfront time to draw it up, then cut them all out using the CNC.
4D