When you say "will the paint adhere to it"...do you really mean paint......or do you mean that you'd be appplying another layer of clear-coat finish?

Which one...will make a huge difference as to what prepping approaches will/might work and won't work.

Yes, I use deglossers and very successfully. No failures to date with paint peeling later on. The best one that I've found is the deglosser sold by Sherwin-Williams. It is HOT stuff. Meaning........it will soften the underlying layer in very short order (5 - 10 seconds)with no rubbing involved. Wipe it on and leave it alone. Cloth damp, not wringing/dripping wet. Cloth...not paper towels is my recommendation. Don't play around rubbing the surface after the liquid is on as the paint will soften enough that it can be pushed around and smeared.

The deglossers sold under the name Savogran are not nearly as hot and will require some rubbing. They never do degloss as well as the SW product, although they seem to do adequately as the paint as never peeled on me when using it. It's just way more work.

Of course, you do not degloss way ahead of yourself/not anymore than you can paint within say 30 minutes.

Depending upon what you have for an exisiting finish (shellac, lacquer, varnish, poly, etc) .........you will likely have adjust your deglosser application technique accordingly. If poly or another type of varnish, a lackadaisical approach will serve okay. If shellac or nitro-lacquer, it will likely melt so fast under the deglosser that you could end up actually removing it if you don't play soft and gentle with your hand and that cloth.

Are you sure you actually want to paint that woodwork? Once covered in paint it will be ten times more work for someone to return to the wood. Often that extra work/money is the deal-breaker,so to speak, and the house never gets returned to its original appearance. For some houses, that's okay. For others...it's a cryin' shame.