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I would make template from 1/4" hardboard the exact
radius of the desired table corner, attach it with double
sided tape, and use a pattern cutting , bearing guided
router bit. The template only needs to be big enough to
support the router base. Just move it from corner to corner.
Your design looks great with the square corners.
The design concept appears to be based on squares so a large round on the corners may look out of place. If you really want a round corner to protect kids heads maybe a gentle round of a 1/2" radius would work and the template method described above would work but you need to be very careful of tear out. I would just use a 1/8" round over bit to ease all the edges
I'd go with Big Jim's suggestion of using a template. You can use any 1/4 or larger stock to make template. I'd pic a modest radius given the design of the table, penciling it in on the template stock, and then cutting it on my little bandsaw. If a bandsaw is not available, you could use a sabre (jig) saw to rouge cut it and then sand it down smooth to meet the layout line.
If you don't want to secure the template with tape (the best way I think), you could make the template large enough to go down the side and end of the table a bit more, and then clamp it in place... basically it needs to be attached in a manner that will ensure it doesn't move. Just be aware that if you're going to clamp it, you need to ensure you're not going to mark the table under the lip, and it needs to be of size where the clamps won't get in the way of the router. (Again, taping is best.)
Then using a bearing guided bit, your edge cut will be exactly the same on each corner.
I would consider not rounding the corners but instead use a rounover bit and run it along the entire edge. If the top is secured already you will have difficulty but you could use a trim router and do the under side too to make a bullnose. I like the look of the table and hard rounded corners may not make it flow as nice.
Thanks for everyone's input. I am just going to use a round over bit on it and leave the corners square. I might sand the corners a little to remove the sharpness of them. I have to make the drawers and hope to post a completed picture of it very soon.