As an alternative to the Star Trek warp travel effect, maybe an Indiana Jones inspired effect could do the trick.
I didn’t have time to work long on this. It could look much better. But you get the idea.
The red line animation is done with Glaxnimate (included in Shotcut). I made a tutorial about this a while ago:
Creating and adding the animation to your driving video involves a few steps. But if you want to give it a try, I’d be happy to guide you trough the process… if you’re not in a hurry 
@Kamigeek suggestion looks very cool though. It’s simpler and has a lot of potential.