Not much in the way of Kingfish around Nelson. You would have to head over to French pass or malbrough sounds to have a decent chance of catching one, having said that its not Impossable to get them around the Nelson area. Ive seen them at The Motueka Fishing platform, Plus hooked and lost one in the moutere Channel.
As for Snapper best spot close to Nelson is the Back beach/Blind Channel. Wasting your time there during the day, But head down there at dusk/Night once all the Dog walkers go home and there is good snapper fishing, fish a couple hours before low tide to a couple of hours after low tide then you will have to leave otherwise you could risk getting trapped by the rising tide.
Other good spots are Ruby bay just fish stright off the beach.
Kina peninsula fish the beach at High tide or the channel at low tide.