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In general objects that appear to us as white either emit a mix of waves with different wavelengths in such a way that we perceive the total of it as roughly equally blue, red and green or reflect all the light that hits them diffusely. So white will always be white since it just means cone triggering equilibrium. Even if your primary colours change.
Fo light emitters it's a bit complicated and partly depends on if our cone cells which are responsible for colour reception would have evolved differently.
With our current sun and atmosphere they have evolved to perceive a range of wavelengths that are the most abundant/intense and don't have a drop in intensity in the middle. Here is a graph showing solar and terrestrial wavelength intensities compared to wavelengths we have evolved to see.
So to find out if the range of wavelengths we are able to see would be different if our star were a red dwarf we would need to take the emission spectrum of the star you'd want to replace our star with(the orange part), then remove from that the percentages of each wavelength that our atmosphere absorbs to get the terrestrial wavelength intensities (the dark blue part).
Then you could probaly look at that graph and take a chunk out of the Y-axis that covers the highest intensity wavelengths (cause plants would probably have that colour and we'd want to see those) while not getting too long and also trying to avoid lower intensity dips in wavelength. Then you've got your visible colour range. If that range is the same as our current one then white always stays white.
However for light emmiting objets depending on if the visible colour range we now perceive is different, our cone cells would also now be triggered at different wavelengths meaning that some stuff that emits roughly an equal amount on each wavelength our cone receptors can perceive which we before saw as white, we would now perceive as colourfull. However all of the natural white light emitters in nature are perceived as such because they are blasting out light on the whole wavelength spectrum basically. So even if our cone cells shifted they'd still be triggered equally and the object would still appear white.
As for the objects that reflect the light diffusely, it would depend on whether they actually absorb some wavelengths that just were outside of our visible wavelength range before. If they do then we would now perceive them as having a colour and if they still diffusely reflect all the wavelengths of now visible light they'd still be white.
Edit: fixed the implications for white perception Edit2:actually answered the questions, structure
There’s also the issue that infrared and UV light is extremely damaging in some cases. Our retina actually can see well into the ultraviolet spectrum, but the lens has a UV filter that blocks anything above violet from passing through. That filter can be overwhelmed though, which is why staring at a black light can be just as painful as staring at a bright lightbulb in the visible spectrum. People who have aphakia (missing the lens in their eye) can see into the UV spectrum.
That UV filter in your lens exists because seeing into the UV spectrum doesn’t offer a large reproductive benefit when compared to its drawbacks. Ultraviolet light is extremely damaging to cells. Especially when those cells are designed specifically to be sensitive to light. Developing retinoblastoma when you’re 8 years old (because the cells in your eyes have been repeatedly damaged by the UV light, and have turned cancerous) means you don’t survive long enough to pass on your genes.
I'd assume if we lived on a world where UV or infrared wavelengths were the most intense we'd evolve eyes to work around the problems that our eyes have. There probably is an upper and lower limit to the wavelengths that animals on earth have adapted to and there probably also is some physical limit (imagine wavelengths of 1 kilometer).
~~However the ability to see in the wavelength that is most intense is a big advantage since everything that does photosynthesis is probably that colour and being able to see those tasty morsels would be an advantage.~~ ~~edit: actually plants reject the wavelengths that are most intense since they are too intense for photosynthesis~~ edit: I have no idea how phosythesis would work on a planet wit a different wavelength distribution.
Edit: thanks for the info though. I didn't know that