When did time go into Warp drive?

As a school boy who grew up watching the original Star Trek series, the term 'Warp Drive' is one old timers like me are familiar with.

The 'Science' in the original show was hazy sometimes, but much of it has become a reality.
Hand held communicators are now iPhones and a reality, powerful personal computers, advanced medical diagnostics, and like devices were only a script writers dream then.
Now they are a part of life.

But the concept of faster than light travel is still just that, a concept and a sci-fi writers dream.
If you want to stretch your brain to the breaking point, ponder the realities of quantum physics, Einstein's theory of Relativity, and is time really linear?

Warp Drive was better explained in later movies as follows;
the universe is folded, and light travels in waves much like sound does.
By finding a way to skip from the top of one wave to the other, great distances can be overcome, allowing travel to distant stars and even to the edge of our galaxy.
Minor matters like momentum, equilibrium, a tiny bit of matter at that speed punching through a Star Ship like a nuclear bomb were never discussed.
Neither was Einstein's and others thoughts on what would happen to the elapse of time if we could exceed the speed of light. Would it speed up, would it reverse or slow?
If a crew left today for Alpha Centauri, when they came back would everyone here be dead?

Personally, one has to suspend logic and just sit back and enjoy shows like Star Trek or Star Wars.
I enjoyed the latest movie that revisits the origins of the crew of the Enterprise.
It was well cast and a real hoot to watch.

But back to my original scatter brained thought from the title of this missive,
just when did the time in my life hit Warp Factor Seven?
I am not now older than my father was when he passed, and older than any of my Uncles were when I was a young adult!
My son is not fixing to be older than I was when I started working full time, and the professional athletes that I watch are now sometimes younger than he is!

Mr Scott, slow this puppy down and put Phasers on stun!