The can of ink is exactly the same for coated or uncoated Pantones. It doesn't matter if you use C or U; it only matters if you ever decide to convert these Pantones to CMYK.
The paper will make the ink react in a different way because of the dot gain; how the paper will absorb the ink. The C or U is just a screen preview to help you during your design process, but the can of ink at the print shop is the same for both.
About dot gain:
Regardless of whether a color is C or U, the ink is made the same.
Note that some printers can modify the ink for you and create a
"custom" ink. But if you decide to go that road, you better not change
printers too often! They will note down your special gray recipe and
use it if you need to get other stuff printed.
Also, a note about printing: if you use a pure Pantone, each press can be manually adjusted as well, so even from one print run to another, there can always be small variations depending on how much the printer cares about consistency, and this can also be slightly different from one printer to another. But it can also help you to adjust your color slightly if you can be present when they start printing your batch of boxes.
Sometimes printers don't even use "real Pantone" inks, they create their own matching recipe. That's understandable and that doesn't mean the colors won't match but that's one part of the reality about "Pantones being always a perfect match". Printing is like cooking, you need fresh inks and fresh paper. For the same reason, not all printers keep in stock all the different kind of inks that have different base. You can search online for ink sellers, you might have hard time finding a "coated" and "uncoated" formula but you might find additives added to the inks to change their viscosity though, inks for different purpose (eg. laser resistant, dries faster) and there's different base for inks as well (eg. soy, oil), as there is for paints.
Will a Pantone Reflex blue look darker on an uncoated paper than a coated one?
Yes and no. Depending on the quality of the printer, a Pantone may look more "saturated" because the ink absorption wasn't properly adjusted to the paper or because your own images were not calibrated for uncoated. If the images are saturated then it could be what you call "darker".
The color might look brighter and "luminous" on most coated stock but that has also a lot to do with the brightness of your paper too. On a very ultra white and bright stock that is uncoated, that Pantone could look better than the same Pantone printed on a low quality coated paper that isn't ultra bright. Same goes for a laminated or a print that has a varnish applied; it might darken a little bit the final result. Since the inks used for offset are not fully opaque, not only the ink absorption will affect the final result but the color of the paper itself will also influence this.
If you are really unsure, the best thing to do is to go at a high quality print shop and ask for samples to compare the results. And ask questions. They know their stocks and inks. You can even ask for a test if you have the budget for it or if they already have the ink in stock. A printed proof will not give you a good rendering of the final result because there's no proofing system that use true Pantone inks. But the print shop can do a test on the stock you chose and that can give you an idea of the final result. If you're using Pantones colors that aren't very popular, it might be hard for you to know exactly how the final result will look like and that's one of the challenge of working with different stocks and inks. You might not get the exact result you have on your screen but the goal is to get as close as possible to it by combining quality of print, stock, brightness, varnishes, etc.
Photoshop has settings to adjust your dot gain or you can do it manually by changing your 100% densities to 85% when you know you're preparing a file for uncoated stock. But before you do this, you might want to discuss with your printer, sometimes they do adjust it as well.
Very good reading from the Mohawk fine papers company:
How to use the Pantone system