I agree, the wallpaper is fun, but, there is way too much of it, so it is also on the point of being gaudy. I would remove it all, and clean the walls, and either put new paper on, just the top half of the walls, or paint it all. (My preference, just paint). Leave the tile alone. They are neutral, and go with the counters, and the wood trim, and the floor. I would leave all the trim alone too.

What are your goals? An-all white, completely plain bathroom? White painted wood?

Painted tile does not last. It will peel very quickly. Focus on the sweat equity of removing the wall paper, then repairing the walls and priming. That will give you a clean slate to SEE that the tile is in good shape, and really not that bad colorwise.

Take all the wallpaper down. Take new pictures of the space and then get back to use.

Let's see the best path for this vintage bathroom.

Is the rest of the home in the same era?

I would try to find a new paper or paint color that worked with the tile.

The wood has gotten a little orange, but probably didn't start this way, but it's possible that I might want to paint some of it.

But really, this bathroom appears in pictures at least to be in such good shape that I would not want to mess with it, and possibly not even redo it.

There's probably not one contractor in four who could build a bathroom with this much longevity and sturdiness these days. You won't be putting back what you take out, quality-wise unless you are willing to spend $$$$

My goals are to update. I’m not afraid of color. The whole home is stuck in 1958 and everything was done really well for that era. I want to be ok with the bones and just modernize it. I like timeless cozy and modern traditional homes.

Well, you can do that either with paint or wallpaper, your choice. I agree, for the era, it was done very well. The tile is actually very nice, and you could easily leave everything but, the wallpaper. I actually saw many that looked similar in 1958.

That wallpaper has to go and I agree remove it then come back for some help.

That tile is pretty awesome! Leave it alone and try to accent it’s vintage charm.

This will be a great bathroom once you remove the wallpaper. I'd paint the walls a creamy grey. Let the wonderful wood and great tiles stay as they are. Change the lighting.

If you want a more subtle paper, this a vinyl faux grasscloth. You could go with something a little stronger if you painted the ceiling instead of papering it.

I have a number of brown and tan, brown and peach variations saved from real estate

You obviously have bought a MCM home in original condition. It looks like great quality. I would suggest you refurbish the interior and not update it. Older homes are not getting the attention they deserve. Everyone want to have a "now" interior. Keep the tile; keep the floor; keep the cabinets (give them a good clean and polish).

The wallpaper...that would be my refurbish project. Although fab for the time, you can find "new" old wallpaper.

Thibaut has a collection of MCM wallpapers, like this one:

Congratulations on the house!!🏠🍾. Since the whole house needs a redo, make a list of priorities and work through it. Don’t waste money on updating something that is functional if you are going to redo it later.

I like the bathroom as is. I do also love color and am not afraid of painting just about anything. Go for it. It's a cheap and easy project. I painted my bathroom floor about 8 years ago. I often see comments about it not lasting but that has not been my experience. Mine still looks great and has required no touchups.

What kind of paint did you use?

The bright whiteness of the sinks and tub indicate they may have been recoated by professional bathroom tub and tile refinishers.

I rolled on regular Behr wall paint and added a stencil on top with chalk paint. I applied several layers of polyurethane for durability. Polyurethane yellows over time but I knew that from the start and was fine with it. I know there are paints specific for this process but this is what I had on hand so it is what I used. I never expected it to last this long. I am very pleased with the outcome and have had no chipping. I would do it again, using the same materials, in a heartbeat.

If you plan on renovating the entire bathroom sometime down the line, the tiles seem to be in pretty good condition. Consequently, I would paint all the cabinets in white rather than paint the tiles. I would install a white baseboards at the floor line, round flush ceiling lights where needed, a contemporary but calm wallpaper, chrome knobs, buy nice bath towels, a white waffle weave shower curtain, a flatweave bath mat. Here's the idea...

Agree with @elcieg - I would redo the wallpaper in a calmer but def MCM vibe and leave everything else (add a neutral sisal looking rug to calm down flooring) - the updated wallpaper will mellow the whole space out and give you time to live with it and make considered & more permanent decisions down the road

FWIW my next door neighbor tore out the original tile in his vintage bath the first summer he moved in and still regrets it 8 yrs later

Do not paint your cabinets white and especially DO NOT install white baseboards as suggested. Leave the tile there. Start by removing the wallpaper from the ceiling and painting the ceiling white or a light color. That would do a lot for this space and might allow you to keep the rest. If not, then you can remove it from the walls too and paint them or find a more neutral wallpaper. Everything looks like it’s in fantastic shape. I love your tile. The wallpaper is kind of fun too. It’s the ceiling that’s too much.

Oh, and I’d remove the current window treatments and outside mount woven wood shades as high above the windows as possible. Choose a shade that picks up on the wood tones in the room or the color of the darker tiles.

Without knowing how much effort and money you want to put into this bathroom, I will suggest the following:

If you know a handyman or you are handy, put flat trim board around the perimeter of all the doors and drawers. This will give the pieces a mission style look. Caulk around the boards and then paint all the wood pieces white. This would include closet doors, entry doors, vanity doors and drawers. Anything wood gets painted.

Take the blind off the window and put in a plantation shutter.

Change the current knobs with modern chrome knobs.

Replace the faucets with modern clean line chrome faucets. Something chunky and simple in design.

Remove the wallpaper. Replace it with a white on white design wallpaper. Or a white textured grass cloth wallpaper. If you can get all the glue off then paint the walls white. Or perhaps a light tan color that matches the wall tile. The walls will look better being a light color.

Use a white cotton shower curtain and white rugs.

If you really love this current wallpaper then keep only a small wall covered. Think accent wall. Perhaps the wall where the toilet sits? You decide. Ceilings should not be covered in wallpaper.

I would remove all the wallpaper and patch up the walls and lime. Wash them with a creamy color. I think that will really elevate the space and you may not have to touch the tile at all.

That is a bathroom that makes me smile! I love (though wouldn't necessarily keep) how they used to be like - we love this wall paper so much we are going to put it on the ceiling and the shower curtain. I bet they had rugs too!). I would not suggest painting tile (I believe it peels off, especially in a moisture rich bathroom). The brown tile with the orange wood is not ideal, but livable if you do plan to renovate in the future. I also would not suggest painting that wood. That wood is valuable. You can't find wood like that in today's cookie cutter homes. If it's the tone that bothers you, you could strip/sand it down; however, I think the orangy-ness can be toned down with the right colors. Removing that wall paper will make a huge difference and then after it's gone, play around with paint colors for the wall that off set any orange wood tones. Changing up the light fixtures could help, new double mirrors over the vanity - depending on how far you want to go with this small remodel. Also, make sure all the light bulbs are cool white, not warm white (to offset orange).

I like timeless cozy and modern traditional homes.

What does that mean exactly?

I echo the comments to tackle the wallpaper first and then reassess. That vintage tilework is spectacular, and painting it makes zero sense. You can strip wallpaper and paint for minimal budget, but dealing with tile is a much larger investment. Painting tile in a bathroom can be done, but if this is a bathroom that will see much use, I'd pick other avenues for updating in the short term.

I am a HUGE fan of vintage tile, but even I feel overwhelmed by your floor tile and would consider changing it out for something period-appropriate but less busy.