Clear[θ]
W[J_,m_,mp_,θ_]:=(∑_(mm=-J)^J)((WignerD[{J,mm,m},θ])^2)KroneckerDelta[mp,mm];
Panel[Column[{
Row[{Slider[Dynamic[J],{0,10,1}],"  J=",Dynamic[J]}],
Row[{Slider[Dynamic[m],{-J,J,1}],"  m=",Dynamic[m]}],
Row[{Slider[Dynamic[θ],{0,π,π/180}],"  θ=",Dynamic[θ*(180/π)//N],Dynamic[BarChart[Table[W[J,m,mp,θ],{mp,-J,J}],ChartLabels->Table[mm,{mm,-J,J}], PlotRange->{0,1},ImageSize->Large,ImagePadding->10]]}]}],"Population of magnetic substates",ImageSize->Full]