in my boat, the galley/salon/vberth is mostly mahogany, but in the head the wood is all teak. its considered a "wet" head, as the shower gets everything wet if you shower in there, which we have only done once, but from others who have used the shower in the past, the teak was, as you said, very tired looking, old and water stained.
after oiling it several times with very little results, I wasnt sure what I was going to do about it.... but then last october i tried lemon pledge, in the spray can.
it is formulated so that it will cling to a vertical surface better than any "liquid" oils will, so I bought a can and sprayed it on... and did NOT wipe it off or try to smooth it out.... i let it set til the next week when i returned to the boat....
when we returnd to the boat the next friday, the oil was gone, the stains were gone, and the wood was a rich brown color. I had found the answer and the problem was solved.
so i thought if one good dose was great, two would be perfect, so I gave it another treatment when I left again... there was some oil to wipe up that ran off and didnt soak in, but the wood seems to be fully oil-saturated now, it looks new, and the water wont cause any harm or staining if we should need to use the shower.
as far as it being a "wet" head, I plan to install a wrap around shower curtain on an overhead track, in an attempt to keep things as dry as possible, which will be a lot better than having to always wipe down the entire head area after showering.
I am going to try the lemon pledge treatment on the cabintop handrails this summer, and see what kind of results we get on exterior wood..
I am not proposing lemon pledge is the secret here, but its the emulsified formulation of it that causes it to cling to the vertical surfaces in a thicker layer is the answer... if you can find another oil that has those properties, i think it should work with equally satisfactory results.