A first aid team and paramedics attended to the 50-year-old for 40 minutes but he did not respond.
Robertson's team was playing at Alex Moore Park in Johnsonville against the North Wellington Zimmers in the over-35s section when the incident occurred.
Cumpstone said Robertson was a family man who strongly supported his family on and off the field.
His daughter Michaela is a rising star in New Zealand football and is involved with the Football Ferns domestic programme.
"She's really well known in the women's scene and her sister Jemma is following in her footsteps, she's near the golden boot in the local league," Cumpstone said.
"He's quite a big personality and is quite vocal in his support for his daughters."
Wellington United said on its Facebook page Robertson would be missed but never forgotten, known as an "energetic, spontaneous, cheeky and loving" family man.
"Even during his own game his mind was on his family... sending messages to see how Michaela and the Diamonds were going. This epitomises Tim and the love that he has for his family and friends.
"Our thoughts, prayers and love are with Tim's wife Lynda, and their children Jemma, Michaela, Holly, Emily, Danielle and Josh, their family, friends and Tim's North Wellington teammates," the post reads.
North Wellington AFC club manager Helen Mallon confirmed his death on the club's Facebook page, acknowledging those who attempted to save him.
"We acknowledge the sense of loss and hurt experienced by the Onslows in particular as well as all those that knew Tim on and off the field.
"Tim was a proud member of the Onslows for over 25 years and will be greatly missed by his friends and family," Mallon said.