In this problem, it is necessary to coordinate the initial and boundary conditions, as well as use the solution method and increase the time interval to 200 for clarity.

eq = { 2423750 Derivative[0, 1][T][x, t] - 
     50  Derivative[2, 0][T][x, t] == 0};
bc = {Derivative[1, 0][T][0, t] == 0, 
   Derivative[1, 0][T][1, 
     t] == (5863/10 - 2*T[1, t]) (1 - Exp[-10 t])};
ic = T[x, 0] == 9463/20;
sol = NDSolveValue[{eq, ic, bc}, T, {x, 0, 1}, {t, 0, 200}, 
  Method -> {"MethodOfLines", 
    "SpatialDiscretization" -> {"TensorProductGrid", 
      "MinPoints" -> 40, "MaxPoints" -> 100, 
      "DifferenceOrder" -> "Pseudospectral"}}, MaxSteps -> 10^6]


{Plot3D[Re[sol[x, t]], {x, 0., 1}, {t, 0, 200}, Mesh -> None, 
  ColorFunction -> Hue, PlotRange -> {450, 500}, 
  AxesLabel -> Automatic], 
 Plot3D[Re[sol[x, t]], {x, 0.9, 1}, {t, 0, 200}, Mesh -> None, 
  ColorFunction -> Hue, PlotRange -> {450, 500}, 
  AxesLabel -> Automatic]}