I'm guessing that if it was made from an old school desk that knees can fit under the cubby.
childrens kness, actually teenagers. Fitting knees under the table is going to be your biggest problem.
Have Mr B sit at your current table and measure up his leg to the knee plus a bit. (thats a precise measurement...knee height plus a bit), but keep in mind the height of the chair will make a difference. If you change chairs you will need to make sure everything still fits.
Can you use your existing dining table?
Fit cross member rails to the legs at each end and run timber between the cross members. Then install the vertical supports. Fix the verticals to the underside of the exiting top using a couple of L brackets. Same L fixings at the bottom of the verticals will save you having to rebate as I have done and then you can use a thinner cut of timber. 15 mm thick should be fine. errrm 1/2 inch? not sure what sizes your timber comes in,
Cross members at the ends might limit the sit-ability of the "head" and "foot" of the table.