The breaking of a wire depends on the stress on the plane of failure. If the stress exceeds the ultimate strength of the wire, it breaks.
initial breaking force (F1)= 200N
initial area (A1) = πr² = A
final area(A2)= π(2r)² = 4πr² = 4A
final breaking force = F2
Since ultimate strength is a material property, it does not change.
So ultimate strength = F1/A1 = F2/A2
or 200/A = F2/4A
or F2= 200*4 = 800N
(in case of any doubt, comment below)