Just to get it out of the way first, "nourishing" wood is a concept invented in the murky past by some marketing team for a company making furniture polish of some kind. The idea that you have to, or even can, "feed" or "nourish" wood is hoo-ha.

Would degreaser spray and furniture wax work?

It could yes, and actually that would generally be a particularly good way to go about it. Specifically, in contrast to using other types of cleaning agents and any kind of oil-based furniture polish, or one that mentions silicone.

Depending on the degreaser however you may want to test it first in an inconspicuous spot. And, again in general, it may be better not to use the degreaser straight from the bottle, but instead dilute it with water; basically it depends on how concentrated it is.

• Before you start, have on hand a small nylon brush for scrubbing (a medium or firm toothbrush is ideal), a few cloth rags and either a towel or abundant paper towels.

• Create a solution of warm or hand-hot water (roughly bathwater temperature, no hotter) and the degreaser.

• Start in the dirtiest areas first. In general the areas that will be dirtiest are contact surfaces and upward-facing surfaces. Use the toothbrush or equivalent for hard-to-reach areas and to scrub out the open grain if there's trapped grime there.

• Clean and dry as you go so that surfaces don't remain wet for too long.

• Swap out the wet rags for fresh ones as needed.

• Once you're done on the cleaning step give the table at least a couple of hours to dry. Half a day or overnight is preferable.

• Then apply the wax as per the instructions, which will generally be to wipe on a generous amount, wait for it to haze over, then buff off all excess. Waxing should be done periodically through the year for furniture that is in use. Once a year or every couple of years is sufficient if the item just sits there and is barely touched, but do dust regularly.