Strong galaxy-galaxy gravitational lensing is the distortion of the paths of light rays from a background galaxy into arcs or rings as viewed from Earth, caused by the gravitational field of an intervening foreground lens galaxy. The vast quantity of strong lenses expected by future large-scale surveys necessitates the development of automated methods to efficiently model their mass profiles. For this purpose, we train an approximate Bayesian convolutional neural network (CNN) to predict mass profile parameters and associated uncertainties, and compare its accuracy to that of conventional parametric modelling. In addition, we present a method for combining the CNN with conventional modelling in an automated fashion, where the CNN provides initial priors on the latter's parameters. These methods are tested on a range of increasingly complex lensing systems, from standard smooth parametric mass and light profiles to images containing hydrodynamical EAGLE galaxies, Hubble Ultra Deep Field source galaxies and the inclusion of foreground mass structures. While the CNN is undoubtedly the fastest modelling method, the combination of the two increases the speed of conventional modelling, especially when priors include CNN-predicted uncertainties. This, combined with significantly improved accuracy, highlights the benefits one can obtain through combining neural networks with conventional techniques in order to achieve an efficient automated modelling approach.