0) ignore stuff like "infinitely quick" since that isn't realistic for what we are doing :-)
1) For the map I am displaying, pick a handful of points of interest to hse as controls, as well as my current location.
2) Use existing data we already have for stuff like direction routing to get travel times between all the pairs in our points of interest.
3) Draw a weighted graph, where each vertex is one of our points of interest and each edge is weighted by travel time. The initial position of each point should be it's geographic position, but we can then use an existing graph layout algorithm that uses an energy-based method (e.g. pretend the edges are springs) to deform the graph. This way each point will end up with its distance being proportional to travel time.
4) Using initial points as our input control points, and their location in the graph as output cobtrol points, use any number of image warping algorithms put there to calculate our warp / morph / deformation field.
5) Apply to initial map image.
I don't know how well it would work in practice but seems like a reasonable first stab at it! :-)
EDIT: Apologies for typo, fat fingers on mobile!