The magnetised Jeffery-Hamel flow due to a point sink or source in convergent and divergent channels is studied. The simplified governing equation ruled by the Reynolds number, the Hartmann number and the divergent-convergent angle with appropriate boundary conditions are solved by the newly proposed Coiflet wavelet-homotopy method. Highly accurate solutions are obtained, whose accuracy is rigidly checked. As compared with the traditional homotopy analysis method, our proposed technique has higher computational efficiency and larger applicable range of physical parameters. Results show that our proposed technique is very convenient to handle strong nonlinear problems without special treatment. It is expected that this technique can be further applied to study complex nonlinear problems in science and engineering involving into extreme physical parameters. Besides, the influence of physically important quantities on the flow is discussed. It is found that wall stretching and shrinking exhibits totally different roles on the flow development. The enhanced Lorenz force affects the flow behaviours significantly for both convergent and divergent cases.