Pc Barker, 44, was directing motorists off the bridge in Workington "saving lives" when it caved in as Cumbria was hit by record rainfall, police said.
Gordon Brown called Pc Barker "very heroic" and the Queen said she was "deeply concerned" about the flooding.
Cumbria Assistant Chief Constable Jerry Graham said both the flooding and the incident involving Pc Barker had left him "devastated".
He said the constable, a father of four whose 45th birthday would have been on Saturday, was a roads policing officer in Workington.
His body was found on a beach in Allonby, almost 10 miles up the coast.
Pc Barker's wife Hazel said her husband was "my best friend, my forever friend, and an amazing dad".
** Our hearts and minds go out to those who are suffering at the loss of a Colleague, a friend, a loved one, most importantly to his 4 children now without a much loved father **