Nickel nuts wear, unsurprisingly, at about the same rate as standard nickel fretwire does. If you're the sort of heavy-handed, heavy-string player who find their frets wear down and need replacing after a few years, you might want to stay clear of nickel nuts. Most players should get at least ten years out of one before it becomes worn enough to require replacing. Players using lighter strings (or anything else that results in less tension, e.g. shorter scale lengths) should get 15+ years out of a nickel nut.
However, a lot of 'nickel' nuts you see around, especially pre-shaped ones, are actually zinc with a thin skin of nickel on top, same as with a lot of cheap bridge saddles. Those wear down much faster. If you go for a nickel nut make sure you buy a blank from a decent company and if a deal seems too good to be true, it likely is. 18% nickel silver is what you want to match fretwire in toughness and tone.
Brass nuts are a little bit softer, but still very tough. They don't wear very noticeably unless you use titanium or steel strings with high tension. Regular dirt common nickel strings are only very slightly harder than brass so wear with them is minimal; pretty much the same as with a nickel nut. If you're Brian May using 24k gold strings then they're actually softer than brass and a brass nut essentially can't be marked by those strings unless you intentionally grind a string down onto it. Even with titanium or steel strings, however, a brass nut should last you ten years of heavy abuse before the slots wear down to the point of needing a replacement. It's not unheard of for brass nuts to get worn down faster, but it's rare. You have to be using very tough strings at very high tension and doing a lot of string bending near the nut to get less than six or seven years out of a nut.
Using a vibrato has no impact on the nut. 'Diving' a vibrato moves the strings vertical to the nut and lowers tension, so you're actually relieving tension there and doing less damage than simply fretting a clean single note does. Pulling back on a floating vibrato does increase tension across the nut, but still no more than bending a note with your hand. Don't worry about a vibrato in regards to nut wear.
FWIW I've still got my first guitar, it's now 16 years old and its plastic nut is still in fine condition even though the frets have worn right down. That's been strung with .010 titanium strings in E Standard for the last ten years. I've got a LP Jr with a brass nut that is now seven years old, using .0105 steel strings and lots of tuning changes (one of those dreaded 'auto-tuning' guitars everybody said would break and be unreliable) and that nut isn't showing a single scratch on it.
Tone-wise, for nickel you can just take any of your current guitars, tune down a half step, then put a capo on the 1st fret. There, that's the tone a nickel nut gives you. Brass is a touch brighter, but not much. (Strangely, brass in any other part of the guitar usually results in a very slightly warmer tone, but for some reason with nuts it always seems to edge brighter by a tiny fraction. I've no idea why, it just seems to always work out that way.) You can also listen to pretty much any guitar which has either a Floyd Rose locking nut or a Wilkinson or LSR roller nut and you'll hear a very similar tone from those.
