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This article investigates the visualizations of heatlines in a natural convection magnetohydrodynamic flow from a vertical cylinder via heat function concept. Fluid is electrically conducting in the existence of an applied magnetic field. The constitutive equations of time-dependent, coupled and highly nonlinear Jeffrey fluid model are evaluated mathematically by utilizing well-organized unconditionally stable finite-difference Crank–Nicolson method. Simulated results are given for several values of Deborah number and Hartmann number to present interesting aspects of the solution of the flow variables, friction factor and heat transfer rate. Results specify that required time to achieve time-independent state rapidly rises with the boosting values of Hartmann number. Boundary-layer flow visualization has been made using heatlines, isotherms and streamlines to perceive the understanding of heat and fluid flow. It is noticed that heat function value reduces for augmenting Hartmann number and also for all smaller physical parameter values and these heatlines become closer to the hot wall. It is also remarked that the hydromagnetic flow-field profiles concerning the Newtonian fluid show a different pattern from that of non-Newtonian Jeffrey fluid. [ABSTRACT FROM AUTHOR]