I run a heavily modded Skyrim (ENB + 120+ mods) on a gaming laptop with 970m and 3GB VRAM. It runs fine. Textures are on high, with a few higher texture mods installed (mostly environmental). I go out of the way to avoid 2k texture mods. VRAM use is usually around 1.8-2.2 GB. But if you want to run on ultra textures with 2k texture mods, then you're gonna want the extra VRAM of the 6GB 1060. Every notch you bump up texture quality increases the VRAM used by textures by 4x.
As mentioned, the bigger problem is with vanilla Skyrim being 32-bit and limited to less than 4GB. The workarounds I've installed to allow all these mods make the game take 30-45 sec to start up, and some game loads can take 30+ seconds on top of that. I don't mind these load times during regular play, but it makes diagnosing problems almost impossible since you often need to restart the game dozens of times as you eliminate one possibility after another. I had to abandon a playthrough I was working on for the last half year because the game would CTD at a certain point, and hours of trying different things (mostly spend staring at loadscreens) couldn't resolve it.
Skyrim Special Edition fixes most of these problems, but the most important mod - Skyrim Script Extender - is only in alpha for SE. Meaning about two dozen of my favorite mods do not work with it.
I haven't noticed Skyrim really stressing the CPU. Granted my laptop is an i7 quad, but I've disabled hyperthreading to keep temps down. CPU use is usually way down near 25%-35%. That said, I do image some of the script-heavy mods would be more reliant on CPU. That said, the 1060 is not that much faster than the 970m, and I'm frequently down to 30 fps on high settings (shadows and grass turned way down). I'm not sure a 1060 is good enough to run it on ultra with an ENB and 2k textures.