The body of Ms Williams, who has a young daughter, was found on a sandbank by another dog walker shortly after 5pm on Wednesday.
Police extended their search after witnesses reported having seen two people walking with "four or five" dogs a short time before. More than 20 volunteers, including fire and local search and rescue crews, took part in the operation, which later recovered Mr Platt's body nearby.
Detective Chief Inspector Mark Pierce, of North Wales Police, said the families of the couple were suffering a "huge amount of distress".
He said: "The two deaths are unexplained but the likely scenario is at the moment that there was unfortunately a tragic accident. We believe the couple were here with dogs and at the time the river was swollen."
Post-mortem examinations are expected to take place on Friday morning to establish the cause of death and a file is being prepared for the coroner.