Don't use exact number since it turns symbolic calculation and probably force `LinearSolve` to get exact solution. Just put for instance `alpha = 45.001 (\[Pi]/180);` and then you get desirable result (I deleted line of code you repeat  twice with `omega = Rectangle[{0, 0}, {a, b}];
mesh = ToElementMesh[omega, MaxCellMeasure -> 0.001];`)

    Needs["NDSolve`FEM`"];
    
    G = 6.894745 10^9;
    
    E1 = 26.25 G; E2 = 1.49 G; G12 = 
     1.04 G; nu12 = 0.28; nu21 = (E2*nu12)/E1;
    
    t = 0.0050 .0254;
    Son = {{1/E1, -nu12/E1, 0}, {-nu21/E2, 1/E2, 0}, {0, 0, 1/G12}}; Qon =
      Inverse[Son];
    
    Q11 = Qon[[1, 1]]; Q12 = Qon[[1, 2]]; Q22 = Qon[[2, 2]]; Q66 = 
     Qon[[3, 3]];
    
    U1 = (3 Q11 + 3 Q22 + 2 Q12 + 4 Q66)/8; U2 = (Q11 - Q22)/
      2; U3 = (Q11 + Q22 - 2 Q12 - 4 Q66)/
      8; U4 = (Q11 + Q22 + 6 Q12 - 4 Q66)/
      8; U5 = (Q11 + Q22 - 2 Q12 + 4 Q66)/8;
    
    alpha = 45.001 (\[Pi]/180);
    a = 1; b = 1; d = a Cos[alpha] + b Sin[alpha];
    omega = Rectangle[{0, 0}, {a, b}];
    mesh = ToElementMesh[omega, MaxCellMeasure -> 0.001, 
       PrecisionGoal -> 5, AccuracyGoal -> 5];
    u0 = 0.01;
    
    angle1 = 10; angle0 = 0;
    angles = {{angle0, angle1}, {-angle0, -angle1}, {angle0, 
        angle1}, {-angle0, -angle1}, {angle0, 
        angle1}, {-angle0, -angle1}, {-angle0, -angle1}, {angle0, 
        angle1}, {-angle0, -angle1}, {angle0, 
        angle1}, {-angle0, -angle1}, {angle0, angle1}};
    
    num = Dimensions[angles][[1]]; h = num*t; pos = Table[0, num + 1];
    pos[[1]] = -h/2;
    For[i = 2, i <= num + 1, i++, pos[[i]] = pos[[i - 1]] + t];
    
    \[Xi]A = {0, 0, 0, 0, 0};
    \[Xi]B = {0, 0, 0, 0, 0};
    \[Xi]D = {0, 0, 0, 0, 0};
    
    For[i = 1, i <= num, i++, T0 = angles[[i, 1]];
      T1 = angles[[i, 2]];
      func[s_] := 
       Simplify@((2.0/d) (T1 - T0) Sqrt[(s - d/2)^2] + T0) (\[Pi]/180);
      theta[x_, y_] := alpha + func[x Cos[alpha] + y Sin[alpha]];
      zA = pos[[i + 1]] - pos[[i]]; zB = pos[[i + 1]]^2 - pos[[i]]^2;
      zD = pos[[i + 1]]^3 - pos[[i]]^3;
      V1 = Cos[2 theta[x, y]]; V2 = Sin[2 theta[x, y]];
      V3 = Cos[4 theta[x, y]]; V4 = Sin[4 theta[x, y]];
      \[Xi]a = {1, V1, V2, V3, V4} zA;
      \[Xi]b = {1, V1, V2, V3, V4} zB;
      \[Xi]d = {1, V1, V2, V3, V4} zD;
      \[Xi]A = \[Xi]A + \[Xi]a;
      \[Xi]B = \[Xi]B + \[Xi]b;
      \[Xi]D = \[Xi]D + \[Xi]d;];
    
    mU = {{U1, U2, 0, U3, 0}, {U4, 0, 0, -U3, 0}, {U1, -U2, 0, U3, 0}, {0,
         0, U2/2, 0, U3}, {0, 0, U2/2, 0, -U3}, {U5, 0, 0, -U3, 0}};
    
    mA = mU.\[Xi]A; mB = (mU.\[Xi]B)/2; mD = (mU.\[Xi]D)/3;
    
    A11[x_, y_] = mA[[1]]; A12[x_, y_] = mA[[2]]; A16[x_, y_] = mA[[4]];
    A22[x_, y_] = mA[[3]]; A26[x_, y_] = mA[[5]]; A66[x_, y_] = mA[[6]];
    D11[x_, y_] = mD[[1]]; D12[x_, y_] = mD[[2]]; D16[x_, y_] = mD[[4]];
    D22[x_, y_] = mD[[3]]; D26[x_, y_] = mD[[5]]; D66[x_, y_] = mD[[6]];
    
    Nx[x_, y_] = 
      A11[x, y] D[u[x, y], {x, 1}] + A12[x, y] D[v[x, y], {y, 1}] + 
       A16[x, y] (D[u[x, y], {y, 1}] + D[v[x, y], {x, 1}]);
    
    Ny[x_, y_] = 
      A12[x, y] D[u[x, y], {x, 1}] + A22[x, y] D[v[x, y], {y, 1}] + 
       A26[x, y] (D[u[x, y], {y, 1}] + D[v[x, y], {x, 1}]);
    
    Nxy[x_, y_] = 
      A16[x, y] D[u[x, y], {x, 1}] + A26[x, y] D[v[x, y], {y, 1}] + 
       A66[x, y] (D[u[x, y], {y, 1}] + D[v[x, y], {x, 1}]);
    
    PDEs = {D[Nx[x, y], {x, 1}] + D[Nxy[x, y], {y, 1}], 
       D[Ny[x, y], {y, 1}] + D[Nxy[x, y], {x, 1}]};
    
   

    gammaD = {DirichletCondition[{v[x, y] == u0, u[x, y] == 0}, y == 0],DirichletCondition[{v[x, y] == -u0, u[x, y] == 0}, y == b]};
            
            
            
            {U, V} = NDSolveValue[{PDEs == {0, 0}, gammaD, 
                Dirichlet
    
    Condition[u[x, y] == 0, x == a/2]}, {u, 
            v}, {x, y} \[Element] mesh]; 

  

 Visualisation 
 

    {DensityPlot[U[x, y], {x, y} \[Element] mesh, 
      ColorFunction -> "Rainbow", PlotLegends -> Automatic], 
     DensityPlot[V[x, y], {x, y} \[Element] mesh, 
      ColorFunction -> "Rainbow", PlotLegends -> Automatic]}  


 [![Figure 1][1]][1]
If we change boundary condition as `gammaD = {DirichletCondition[{v[x, y] == u0}, y == 0], 
   DirichletCondition[{v[x, y] == -u0}, y == b]};`, then the solution looks as follows 
 

[![Figure 2][2]][2]


  [1]: https://i.sstatic.net/HmJPe.png
  [2]: https://i.sstatic.net/GMbm2.png