To more directly answer the question asked, the necks with the larger nuts will not feel thinner, but they will feel flatter. The curve from the middle of the neck to the edge is gentler on wider necks, as it's spread further. This is the reverse of how fretboard radius feel more and more curved the wider the neck is.
If you're familiar with the idea of neck profiles being in a 'V', 'C', 'D', or 'U' shape, you can think of it this way: the smaller the neck is the closer to 'C' and the wider it is the more you move to 'U'.
Hence, the 'standard thin' Warmoth profile with a 1 11/16 nut is nearly identical to the standard thin U ESP neck shape, but with a 1 5/8 nut it is closer to Epiphone's 'D' profile. (Note: I completely refute the idea that Warmoth's 'standard thin' is ever like a Fender 'C' neck, 'cause you only have to measure 'em to see that they are well off, let alone actually handle one.)
(As far as Warmoth goes, only the 'boatneck' and 'Clapton' approach a 'V' [and they're both very soft 'V's])
For the 'fatback', it's the same deal. The narrower the nut the closer it comes to a 'C' and the wider you go the more of a 'U' it is. So if you like a flatback and high shoulders, go for the wider nuts. If you like a fully rounded neck, go for one of the narrower nuts.
As a general rule, rounder necks fill your hand more if you play 'thumb over' or 'palm' the neck, while flatter necks will feel more substantial if you keep just your thumb on the back of the neck and your hand below. But how much either 'fills' your hand also depends greatly on the size and shape of your hands, so it's not really something anybody can guess for you. Your best bet is to measure your existing guitars to get a better idea of what you might like; failing that, just think about the C/D/U shape, and go with whichever of those you think you'll like most.
