And camera flashes are covered with UV-filtering plastic, making the effect even more negligible. So in essence, no, it's practically not a problem.
True. I've just checked that plastic on my oldest camera and there's absolutely no degradation (such as yellowing) present. And this plastic is just about 1 mm away from the flash tube. Some discharge lamps (especially MV/MBF but probably also MH ones) cause plastics yellowing due to UV /B(?) leaks through the outer bulb glass. So the UV is indeed an insignificant problem here.
For the UV exposure: There are shown different pieces, each of them sensitive in a different manner.
The fixed lighting installation in the museum could be designed to take these specific limitations into account, so those items not sensitive at all are placed in areas lit even by a natural daylight, but with those most sensitive ones you won't find any fluorescent nor HID light source.
That's a good explanation. Incandescent (halogen) spotlights might be a problem due to overheating the exhibit and (if not glass shielded) by to adding a tiny bit of UV, too. Today, I assume LED spotlights would be the best option here.
Why do they prohibit any photographing (even without the flash) might be also because so many people can't control their cameras well, use them in the point-and-shoot mode and have no control over the flash, letting the camera decide.
Btw. I've also visited a few places (e.g. the "Skanzen" in Stockholm) where they explicitly told us when visiting an indoor exhibition where spinning textiles using original equipment took place - taking pictures OK but please no flash! Obviously for the discolouring reasons listed here.