The Saffman-Taylor (S-T) instability mechanism in laminar premixed flames in a Hele-Shaw cell is investigated using two-dimensional numerical simulations with an Arrhenius reaction model and Poiseuille assumption for the viscous effect. The baseline calculations considering the Darrieus-Landau (D-L) and diffusive-thermal instability modes show results consistent with the classical linear instability theory. The primary effect of the variable transport properties is found to be the modification of the flame thickness, such that the results can be properly normalized by the actual flame thickness and timescales. The effect of different Lewis numbers is also found to be consistent with previous studies. With the S-T instability mechanism, the overall effect is to enhance the destabilizing mechanism by providing an increased viscous force in the product gas. The linear instability behaviour is found to be qualitatively similar to the D-L mechanism. However, the results in the nonlinear range demonstrate that there may exist distinct characteristic timescales associated with D-L and S-T mechanisms, such that the latter effect sustains longer in time, contributing to a higher overall flame speed. The calculations show that the S-T effect is considerable for Peclet numbers less than 50. For sufficiently smaller Peclet numbers, the overall flame speed is found to be significantly affected by the S-T mechanism.