Workaround: Saving Problems
There is at least one reliable workaround to these problems (which are very annoying for both you and us, as it's really hard to track what's going on). I have not lost a single car since the recent update and it seems to be the following procedure that ensures that.
Building a new model, first trim:
Build a rough version of your car to completion first, test it on the final testing page.
Go back into the engine designer, final testing tab. Name the engine family and variant. Hit the save button. (This step might not be necessary)
Go to the final testing tab and name your model and trim. Hit the save button.
Continue to modify the car to your liking; before quitting, go to the final testing page and hit save again.
Building a trim after the first:
Clone your first trim using the clone function in the drop-down menu.
Select this trim, but first name it when on the final testing page after updating all the aspects you want. Hit the save button.
Continue to modify the car to your liking; before quitting, go to the final testing page and hit save again.
As a rule of thumb: only name a model, trim, family or variant after the whole thing is done in its first iteration. Don't bother designing it visually in the first go, just finish a rough draft before putting more "work" in. Once the rough draft is done and saved, just modify it as you like, always end with going to the final testing page (of engine designer for engines, of car designer for cars) and hit the save button.
Note: Unfortunately this does not work for cars that are experiencing the bug on loading already.
Hope that helps, cheers!