From the photos the floor tile, toilet and tub appear to be in good shape. A new countertop, sink and faucets (if needed) and an undermount sink would go a long way to updating., as well as cabinet doors, then refresh the linen closet doors. Kitchen cabinet height is popular now if you want to replace the vanity, but then you have to move the mirror and maybe would affect the beautiful tile floor. If the tub needs reglazing go for it.
New window treatment - maybe top-down/bottom-up cellular shades - to control light and airflow. Wallpaper would add some color and pattern. Bright colors would accent the tile.

Here is a simple solution for older bathrooms without a niche for bath products in the tub. Big thank you to my son who came up with this idea. Make sure your hook extends beyond the top edge of the tile or tub surround and use wall anchors if there is no stud. This is our guest bath so only a small unit was required, but he installed a larger one for the family use bathroom in his house. So nice to have smart kids 😉

Your bathroom is fine but blah and too dark. Here's some basics ways to upgrade, keeping everything in place.

--Paint bathroom walls white with a touch of pink. Benjamin Moore Atrium White is great for bathrooms.

--Paint the vanity base black and get more modern hardware. Coordinate all hardware you decide on--black, nickel, brass--towel bars, lights, faucets, etc. Add new white or lightly patterned countertop.

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--Liberate the bathtub from the frame and add either a plain white shower.

--Get a new wood trimmed mirror

--Add wood shelves over the toilet.

--Add rattan shade over the window

I see no big things about the bathroom , no way do you ever refinish a tub it does not last and if you think it is bad now just wait until it starts peeling. Gey a nicer brighter light and if you have a color you like in the bedroom next to this paint the vanity thta color . Anything else will be expensive save the money until you can do it right .

What is your budget? How long do you expect to live in this house? Does everything work well? Any leaks? Is this your master bath?

I think it looks fine, but if it were me, and everything worked well, and I planned to stay in the house for a long time, I'd save up for a total gut. Which is what I've done in my forever home. All the bathrooms have been totally renovated but we lived in the house for quite a few years before we did them.

For now you could install wallpaper, get a new window treatment, and try some new colorful accessories. Anything more than that is going to be effort and money that you could save for a future dream bath.

I would lighten the walls and bring colour in the accessories. If you can replace the vanity doors and drawer fronts that would be great but if not simply paint everything white and install new hardware. A low vessel sink would update the look of the countertop with your white tiles and is easy to install with a new faucet. If needed, install chrome or polished finish screw-in or adhesive caddies for the shower/bathtub. You can also replace the closet louvre doors with bi-fold in the same style as the other door.

Looking at the tub drain combo your bath probably from the 60’s. A caution, I wanted to replace drain system but plumber cautioned all sorts of unknown difficulties with replacing the tub drain. That would have to wait for a gut job. Something not doing as never used. Installed few years back the same floor tiles. Classic. As suggested above, I would put dollars to a new counter and sink with new faucets if needed. Then paint the vanity a fresher white, a color or black, new hardware including towel rods, an interesting mirror. Add art work over toilet and a pretty but washable smallish rug. Paint walls a pale color to brighten the space. Some sort of window covering for bottom so natural light comes in and easy to open window for fresh air. If can, new linen closet doors would really finish the bath. A weekend of work plus countertops installation. Result, reasonable dollars, a new bath.

Wow, it looks really nice--and I love the dark green walls!

Looks great! New towels for the photo shoot!