Can't say positively but the odds are the inner part, spigot, is simply screwed to the wall. THe spigot goes through the wall to the outside and is part of the inner ring you see.
The caulking will still hold it quite firmly even if the caulk is leaking.
The outside wood trim most likely covers the exterior trim plate which is also screwed to the wall, just on a different pattern so the screws don't conflict with each other.
Remove the wood, remove the exterior trim ring and then you can work on the spigot which is part of the interior trim plate/ring.
You will need to cut the caulk with a variety of tools. Try knives, chisels. The Oscillating tools with toothed blades or the caulk cutter with no teeth.
Sometimes carefull use of heat for a heat gun or torch can help the caulk release. Just be carefull to not overheat and scorch the wall.
I would also remove the flip up glass setting it aside with all its hardware. These are likely offshore made and hardware losses or breakages could be problematic.