The 25-year-old agreed to join the Tricky Trees over the weeked, bringing an end to his three-year association with West Brom.
Having sanctioned the sale of the striker, Ashworth has praised the Republic of Ireland international for the way that he conducted himself during his time at the Hawthorns.
"I would like to place on record my thanks to Coxy for his great service to the club," he told the club's official website. "He has a fantastic attitude which I think is best summed up by the fact he did not miss a single game through illness or injury.
"We've had lots of interest in him this summer and this is a good move for him and his career.
"He joins Nottingham Forest with our best wishes."
Cox is expected to make his Forest debut during Saturday's Championship opener against Bristol City.