<p>This paper proposes two different Taylor polynomial-based numerical algorithms, one using uniform collocation points and the other using non-uniform collocation points for the numerical solution of a class of third-order Emden–Fowler type pantograph differential equations. The sufficient conditions for a unique solution to the problem are also derived for the first time. In the literature, only the weakly nonlinear IVP of the concerned model is tackled. In the present work, BVPs and strongly nonlinear IVPs are also considered. Both algorithms are designed by employing the operational matrices of the delayed-derivative terms appearing in the model, accelerating computation. The collocation schemes are appropriately designed to handle singularity effectively without removing it. This also reduces the considered problems into a nonlinear system of equations, which can be solved by the Newton–Raphson iterative method. The accuracy of the algorithms is further analyzed by some theoretical error bounds for the uniform and non-uniform collocation approaches. Several numerical illustrations are provided to support our investigation. In addition, two singularly perturbed problems are also solved to check the flexibility of the proposed algorithms. The fast Taylor operational matrix algorithms tackle several singular differential problems, including perturbed ones, effectively. Also, the numerical outcomes of both the proposed algorithms are compared to decide which is more efficient for the respective problems.</p>