The native of Hamilton, Ontario was drafted 27th overall by the Montreal Canadiens in the 1980 NHL Entry Draft after a fine rookie season with the Brantford Alexanders of the OHA. He was returned to junior and registered 84 assists in two years while playing a hard-nosed style behind his own blueline.

"Stash" made his pro debut by playing five Calder Cup playoff games for the Nova Scotia Voyageurs in 1982. He played 40 games for the rebuilding Habs in 1982-83 but his role was reduced over the next two seasons. In addition, he served a 30-game suspension after being charged with possession of marijuana and hashish.

Nattress' career was given a boost just prior to the 1985-86 season when he was traded to the St. Louis Blues for cash. His defensive savvy clicked with the style of play favoured by coach Jacques Demers and the Blues reached the semi-finals for the first time in 14 years. The next year he teamed with Jim Pavese and was a top penalty killer as St. Louis finished first in the Norris Division before being upset in the first round by the Toronto Maple Leafs.

In June 1987, the veteran rearguard was traded to the Calgary Flames where played a steady defensive role on one of the league's top teams. The Flames topped the NHL in points during the 1987-88 season then won it all the next year in a six-game final versus Montreal. Nattress continued to be a stalwart on the club's blueline and represented Canada at the 1991 World Championships.

Part way through the 1991-92 season, he was traded to the Maple Leafs in the deal that saw centre Doug Gilmour and veteran Jamie Macoun head east and Gary Leeman as the key figure heading to the Flames. Nattress provided stability to the previously weak Toronto defence and helped the team come close to making the post-season. In the off-season he decided to explore his options and signed as a free agent with the Philadelphia Flyers. He played 44 games in Philly before calling it quits after the 1992-93 season.