Transient conjugated heat transfer in simultaneously developing laminar pipe flow is analyzed involving two-dimensional wall and fluid axial conduction. The problem is solved numerically by a finite difference method in a thick walled semi-infinite pipe which is initially isothermal, with hydrodynamically and thermally developing flow and with a sudden change in the ambient temperature. A parametric study is done to analyze the effects of six parameters, namely, wall thickness ratio, wall-to-fluid thermal conductivity ratio, wall-to-fluid thermal diffusivity ratio, the Peclet number, the Biot number and the Prandtl number. Heat transfer characteristics are seen to be highly affected by the parameter values.