Well I made the research I should have made in the first place

LED lamps are, indeed, UV, mostly UV-A. It's UV light that
cures the gel varnish so it becomes solid. The varnish contains some molecules called photo initiators that do the magic. The fastest photo initiators have a orange-ish tint, which is an issue for the clearer, lighter pigmented colours. For these, different photo initiators have to be used, but these are hardly activated by light frequencies within UV-A.
As you said, LEDs emit a narrower band, totally contained within the UV-A range. It causes smaller times for the varnishes that cure at that frequency range but takes longer for others, especially the lighter colours. The bulbs can live for up to 50k hours and need not be replaced. It makes most sense to get the lamp from the same brand as the varnishes because that reduces the curing time to the minimum.
Non-LED lamps (CFL—compact fluorescent lamp) have a broader frequency range with a less pronounced peak. Takes longer for the gel to cure, but lighter colours are an easier target. The bulbs last for ~10k hours and need replacement after 2-4 months depending on how heavily they are used.