After Dougie Brown, with 60, and Brian Lara, who hit a rapid 49, had provided the bulk of Warwickshire's runs, the West Indian all-rounder Simmons came up with his vital knock, which included five fours and four sixes.
After Simmons and Vince Wells had put on 73 for the fourth wicket, it was left to Nixon to hit the winning runs having made 36 off 33 balls.
The Zimbabwean Test bowler Paul Strang claimed 4 for 49 as Nottinghamshire beat Northamptonshire by 35 runs at Wantage Road to maintain their hopes of a quarter-final place from a tight Group A.
Strang helped restrict the hosts to 237 all out after Nottinghamshire had made 272 for 7 off their 50 overs. Graeme Archer cracked 70 off 71 balls to bring up the winning total.
Gloucestershire logged their first Group C win as they beat Somerset by 26 runs in the West Country derby at Bristol.
Tony Wright, with 44, led the home side's charge as they reached 234 for 7 before four bowlers each took two wickets to assure victory with two overs to spare.Reuse content