I think, white LED bulbs do use a single wavelength blue LED, but they pass it through phosphors which transform it into a similar distribution of wavelength that you would find from an incandescent bulb, or sunlight. LED's are a lot better than fluorescent lighting, which does often have very dramatic peaks and gaps.

In theory, you can look at the CRI rating, but in fact CRI only uses a small number of wavelength probes for the test, so a rating of 100 doesn't necessarily mean a smooth spectrum, it could be the manufacturer has tuned wavelength peaks in order to game the test. There are better rating systems being developed, but they're not used by manufacturers or mandated by governments yet.

I actually think it might be the lightbulbs that change colour which could be problematic. I wouldn't be suprised if some of those did generate the colour just with single wavelength blue, green and red sources. Hopefully the better ones would have individual phosphors for each r/g/b source which produces a spectrum which when combined is reasonable.

I would be particularly careful about cheap/unbranded colour-variable lightbulbs. I doubt they've been through much or any consumer testing, it seems you can buy anything straight off the boat from China through domestic web stores these days.