Some optical glass is transparent to UV. Some is not. And of those that are transparent to UV they may not transmit enough, or over a wide enough range, to make a noticeable difference in many use cases.
The brightness difference between Visible+UV and Visible-UV may not not have much of an impact on the image, and so far I've considered there increase flare issues of an additional filter to outweigh the downsides of slight UV exposure with my gear.
I have some negatives kicking around from last summer where I shot a few rolls (delta 100) with different filter combinations on my Mamiya TLR 80mm lens one afternoon down on the waterfront. Bare Lens, UV Filter, Red Filter and UV at the same time, Red Filter only, etc. Overall effect of the UV filter on the final image compared to the others seemed to be not worth my effort. The UV filter vs bare lens had a bit of contrast improvement, but any effect the UV filter had when paired with any of my coloured filters seemed negligible.
The UV filter I had on hand at the time however was a rather cheap one and possibly of questionable value. I've honestly been on the fence about spending the money to experiment with a better filter to see if it makes any kind of a difference.
Sadly digging up reliable specifications on different lenses and filters for details on UV transmission profiles seems terribly hit and miss if you're trying to quickly google things. (The other 'fun' thing about UV transmission is that it can vary on manufacture. I would need to go digging again to find it, but I do remember something about one type of glass was horribly random for UV, swinging from blocking nearly all to nearly nothing, depending on factors like how quickly the glass was cooled when being made. This made for some 'fun' results during some science experiments back in the day when UV was still not well known.)
But if you are planning to begin experimenting with additional filters, then you may want to go ahead and spend the bit of extra film doing your own comparison shots. I feel that one of the best things I've done to help get an understanding of the different filters impacts on an image has been taking my own shots with various combinations of a scene I've been looking at with my own eyes. I went in with an idea of what was probably going to happen, but the reinforcement of doing it for myself made a nice confidence boost.
Just remember to keep good notes on which filters you're using in which frame

I need to dig up my test photos from last summer and put them together as an article at some point, but I might invest in the project with a few more filters and just redo all the shots.