Personally I always CA finish after engraving and color fill is done. This way the color wont chip out or get dirty. The CA finish seals it all and will be perfectly smooth which makes for a nice feel.
I turn the body slightly smaller than the required size before it goes to engraver, then build up above bushing size slightly with CA. Then sand and polish to required diameter.
On wood I usually finish before engraving. I find that the laser will engrave too hot, too deep etc - especially on softer woods, spalted, burls - but with the CA soaked in and finished the wood becomes more consistent and I get a more consistent engraving. I sometimes add CA after engraving to protect filler paint if used but on wood I find it doesn't usually need it.
I finish with shellac or other friction polish first, which I suspect would distort the light ashy lettering if burned with the low end lasers like the 1500mw Neje style laser I use. The 40 and greater watt lasers can produce deeper carving rather then just top burning, so likely not an issue if the laser burns lower than the level of the turned wood. If by repeat engraving with low end laser the name is lower than the surrounding wood, that would be a great balance, but then the friction polish would accumulate in the depression of the name...I'll experiment if no one already knows...
I have the 1W Neje and always finish with CA before engraving. I tried it with and without finishing and found I get a cleaner more precise burn with the finish on. With the higher power settings I get a deep enough burn I can color fill if I want.