I wouldn't do it. It's inevitable that you will get water on it, no matter how careful you are. You can dry it off but I think over time it will take its toll on the wood finish. I can't imagine that having to refinish a wood surface around a tub would be all that easy.

We have quite a bit of cherry wood around our jetted tub, including on the vertical surfaces at the ends of the tub. It does get splashed occasionally so I keep a small towel by the tub when it's being used so that I can immediately dry everything off.

If a marble slab is not in the budget, I'd look for a large format tile that you could use and then be sure to waterproof the deck surround since the grout is not.

Why would having a wood surround for a tub be any different from having a wood counter top for the sinks? There are people that have their sinks in wood counter tops. I know there are some over on the kitchen forum, perhaps you could ask them how they deal with splashing.

While I haven't done exactly that, if you wanted to do it check out the marine wood finishes for boats. I would expect that to be very hardy.

I have a wood tub deck, but it's teak.

There's been a lot of splish splash but zero issues. With a good film finish and simply wiping up any standing water after you use the tub, I'd see no honest difference between the two woods in terms of wear and tear.

I think with the proper finish and upkeep it would be fine.
(We dry all the tile and the top edge of the tub after every shower, and it takes much less than a minute, it's easy).

It would also be historically appropriate because the first bathrooms in houses like this were often box tubs, and the tub and toilet with both encased in wood boxes or cabinets. A house near me has a 19th century bathroom that is all wood with a metal tub basin. The plumbing is so old it is all external and looks like a series of garden spigots.

I know you are not going "historical" in the bathrooms but it would make a reference to history.