Cleaning off urine from anything is a royal PITA. Cat spray and dog urine are also bad.
If the urine has sat on the tables long enough to mold them, you definitely have problems.
I've had to "clean up" a number of pet related messes on furniture over the years as well as a few rodent messes on some older pieces.
Fresh urine (and some older stuff - if it still has urine smell):
Lots of a product called "Nature's Miracle Urine Destroyer". Follow their procedure.
Follow it with Murphy's Oil Soap.
After numerous washing - refinish as needed.
For mold - that's a tough one. Chlorine bleach will effectively kill the mold if any is left - but may not lift the stain out of the wood. Concentrated hydrogen peroxide will lift the stain out of the wood but will also bleach the color of the wood - so if it is cherry or walnut, it would be "decolorized". I would try the chlorine bleach first.
Try it is some inconspicuous places and see what happens to your wood.
As far as stripping the wood stain from your finished pieces - many of the commercial strippers work well.
Your choice as to what you want to use. I've used the methylene choride based strippers as well as "Citri-strip". Both work reasonably well.