If I were in your shoes I would replace the beadboard in the tub area with tile. A shower means splashing water, which means it will hit the walls and you can't avoid it. There isn't a way to waterproof the beadboard, so I would replace it.

I know that isn't what you wanted to do, though.

As for actually fitting a shower in there, you need to reverse the tub so the plumbing is on the tall wall (same wall that the sink is on) and with that shape of tub you'd probably have to replace it to make it work.

I don't think a minimal reno will let you achieve a shower in this space ... do you need a shower on this floor? What does the rest of the floor plan look like for this bathroom?

Can you waterproof the beadboard with marine varnish? Then replace the tub filler with a combination thermostatic shower valve/tub filler, attaching a hose and handheld shower to the tub filler (such as the Grohe Grohtherm 3000), and a shower bar to allow you to adjust adjust the height of the shower head. Add a bar for the shower curtain.

You've got unused attic space on the soap dish wall, that you could expand into? Do you want to keep the tub or replace it with a shower stall?

Anyway, the simplest way to get a shower in is to do as Pam says, putting the faucet set on the other end of the tub and tiling the walls. White 4x4 tile would be true to the period, you could keep the beadboard ceiling but you would have to make sure you had good ventilation to take care of the extra steam a shower makes.

And you would then have the drain at the other end of the tub than the faucet, which I have never seen.

My brother tiled the tub surround in my grandmother's 1950 cape so people could shower. It works with the vintage style. You could save the removed beadboard for another use.

Thanks for the feedback and ideas. The varnish idea is intriguing; I think I've seen similar applications and they end up looking very shiny and not a match for the paint. This is not so terrible, but would be a bit of a different look. Are there any varnish-type products which would have a more neutral finish? Also do you think I'd need to varnish the ceiling as well to protect from condensation, or only on the walls which would be receiving direct 'splashes'?

We are leaning towards keeping the tub, and retrofitting it with a shower. The idea of a hand-held European-style shower is very appealing; we're wondering if maybe a ceiling-mounted curtain ring would be sufficient to prevent splashes if the shower hardware were somehow mounted inside the curtain, maybe from a bar mounted to the ceiling? We did something similar to a clawfoot tub in our previous house - link added to show what that may look like. It would be a tricky install since the curtain ring would probably need to be mounted at the same angle as the slope of the ceiling. We would probably just leave the current tub faucet set in place and add another faucet set that operates just the shower.

There's not really an economical way to create space for a separate shower, changing the footprint of the room is off-the-table for us at this point.

Here is a link that might be useful: clawfoot shower system

Varnish won't do anything to keep water out from between the boards.

As far as a curtain goes, you could cut the top at a diagonal to follow your ceiling line, and put large grommets in it, hooks/pegs on the wall to hang it from. A curtain rod where the ceiling is level, and then the hooks where the ceiling drops.

I am very afraid of mixing wood and water. I wouldn't want a shower or tub around wood. Too much direct contact with water. Rot and mold would be my concern.

Can't you replumb the wall that currently house the faucet and pipes now such that a shower head comes out of the sloped area? Sort of like a rainshower head?

I'd remove the beadboard around the tub and tile the walls after I made sure that it was well waterproofed behind the walls with a plastic/membrane followed by cement board. And I'd make sure that there was a strong bathroom fan in place to take care of any humidity in the room to protect the rest of the beadboarded walls.

You can find beautiful tile that would complement the rest of the beadboarding in the room.

We used a curtain-ring kit like that on a claw-foot tub we wanted to keep, at the same time as wanting a shower. It worked well. Clear shower curtains (I think they were actually curtain liners but nice and heavy) let in lots of light and weren't intrusive. IIRC, we needed three standard curtains to encircle the shower, overlapping them for two or three rings at the seams.

I think that even if a shower curtain on a ring protects the walls, you'll still have a lot of water splashing onto the sloped ceiling which will run down the grooves & into/through the gap between the ceiling beadboard & the wall. Tiling the tub alcove would be a good idea, and it would give you access to the plumbing in the wall (which means it would be easier to add the new shower fixture before you button everything back up again).

I have a 126 year old lakehouse for the summer months and I have beadboard throughout the tiny bathroom. It has a claw foot tub and we put a surround type vintage looking shower curtain. I slide it open to the front right corner all day, but if one wants to take a show, then wrap it around the tub. We've had the house for 17 years and the beadboard is as good as they day it was installed almost 45 years ago. So yes you can do that and maintain the footprint and avoid any deconstruction. I love the looks of it.