Pics of the bathroom and master bedroom will help as the door, window and base trim all needs to somewhat tie together.

I have done both. Going with wood is the easier choice, specially if it is painted white to match all the white trim in the house.

I agree with @chispa.

It is limiting to find a bathroom floor tile that has a matching baseboard tile, and expensive to have a tile-setter cut the tile for a baseboard when there is no matching bullnose.

I used a bullnose tile as the baseboard in two bathrooms (10 years ago) and found that dust sticks to that 3/8-inch thick rounded top edge on the tile. Also, the tile baseboard does not seem to line up with the door casing next to it.

I'm returning to painted wood for the baseboard in my next bathroom remodel. I plan to match the height to the baseboards in the living spaces, 6 inches.

What size are your floor tiles? Plain or patterned? Can you show us your floor tile? If larger plain tiles, use those for the baseboard. You don’t have to use bull nose. Can be cut/squared off, and your tiler can use a metal strip on the top edge. There is a name for that strip, but I can’t remember what it is! Your tiler will know…. I have those in two of our bathrooms and have held up really well - just have to run a rag along the top edge to clean off any dust that collects, but no more work than dusting a baseboard. That will give you a more contemporary look.