In this paper, a new functional link neural network based on Lagrange polynomials is introduced. Due to the well-known efficiency of Lagrange polynomials for solving different problems in fractional calculus, these functions are considered as the neuro-solutions for a class of nonlinear variable-order fractional time-varying delay differential equations. Compared with the other kinds of neural networks such as multilayer perceptron, the suggested functional link neural network shows more satisfactory results. Since the order value of derivative and delay are not constant, it is crucial to choose an appropriate activation function for solving these problems. Then, the effect of some common important activation functions on the numerical simulations is studied. The results for several examples are examined that confirm the performance of the proposed functional neural network.