Means smart, stylish, and properly well turned out. If someone’s natty, they’ve made an effort and it shows, like sharp suit, tidy shoes, hair actually behaving for once. It’s a bit old school, slightly posh, and usually said as a compliment, not sarcasm. Honestly, it deserves a comeback.
"Rolled into the pub in a fresh suit and polished shoes, looking proper natty. Dave goes, Oi, who nicked you from Savile Row? Even the barmaid gave me a nod."