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I have some cherry left over from a bass I made and noticed that there's just enough wood for a soprano sized Uke to be made out of it. Problem is the the grain of the wood. Since the instrument is so small and there's (comparably) little tension, is the structural integrity compromised by going against the grain?

It's not so much the structural integrity that's the problem (although that could be a problem too), but to make the top stiff enough, you'd have to make it way thicker than normal which would be very bad for the sound. Wood is far stiffer along the grain than across the grain.
It may just about be usable for the back.

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Humidity expansion could be an issue as well. The back lengthening and shortening with humidity changes could affect the neck angle. But I'd say it would probably be ok for a back if you put two or 3 braces going from head to tail rather than the usual side to side style.

I'm sure it would be possible to make it survive as a top as well, but it probably wouldn't sound as good as the usual alignment. Could be an interesting experiment though... sort of like the opposite of ladder bracing

Totally spitballing, but maybe you could brace it with two "parallel" (actually divergent but longitudinal) braces from neck to tail block in the style of an archtop guitar? You would have to use f-holes, or at least some non standard sound hole. Come to think of it, I guess I'm describing a mandolin. Oh well... might work.

Cut it out and make the back with the necessary bracing and set it aside for a while , lets see what happens . You have asked a question that needs a bit of experimentation . What do you have to lose ??

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I'm not sure the stiffness across the grain (or lack there of) is as much of a concern as you might guess. Softwoods have a much bigger difference in stiffness (across the grain compared to along the grain) than hardwoods do. Further, we are talking about a very short body. I would say try it out as an experiment and see what happens. Alter the bracing using your best judgement and educated guesses and see what happens.

The real question is, what are your expectations for this instrument? If you want a nice uke and don't want to assume the risk that it will be a bust, then you can certainly find a suitable plate in soprano uke size elsewhere. If you are looking for a fun way to use up this scrap and maybe learn something along the way, go for it.

I designed my first couple instruments as much around the wood I had as I did around what I wanted it to be. That was fun. Once I designed an instrument where the primary design factor was the resaw capacity of my little bench top bandsaw. Everything was designed around that maximum body width. Though I am glad I have a bigger saw now and have built up a tonewood stash. . .

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I can't see all of your stock in the photo but it looks like you have enough to try a two piece back with the proper grain direction. It wont be book matched but it should look OK. That board looks like a beaver has been chewing on it!

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It all depends what you want from your uke. There are two approaches:

1. Make it like a guitar with a softwood top. In this case you can use stiff back and sides, and that wood will be fine for a back with 2 x 4 longitudinal bracing (shaved down a little perhaps). I really don't like this style for sopranos (well, not at all really, for any size - if you want a guitar make a guitar ...), because they almost always sound shrill and "scratchy". But some love 'em.

2. Make it more traditionally from all hardwood. In this case you really have to build light as you can, otherwise you will produce a near-silent uke. But if you get it right, they sound really good. My aiming point for a soprano is 240-300 grams (8.5 to 10.5 ounces, including strings and hardware). I couldn't use that wood because it wouldn't be stiff enough along the long axis, even if you left it too thick to make much of a sound. The big split into the upper bout won't help either.

Cherry can make very fine soprano ukes, but not that piece for my taste. I'd buy some more cherry and use that piece for headplates.

If you build with cross-grain top, consider "A" bracing or falcate bracing. That will improve the longitudinal strength significantly. The string tension on a uke (tenor) is only around forty pounds so if your linings are sound you should be OK.

Bob

Bryan Bear has it. You probably don't want to hear his answer, but listen to it anyway. You can buy a proper uke top way cheap. Do that. In the long run, you won't be sorry.

...or...pursue your path with orienting the grain of your current "top" in the wrong axis. Report back to us. I think you will be disappointed but I am often wrong. Just report back to us honestly, either way.