There seem to be two parts, or "stages," to the answer here.
When and why was "down the road" used to refer to the future in a figurative way?
When and why did this expression become idiomatic in English, recognizable and commonly used in colloquial speech and processed as an idiom.
The difference between the two is that in stage 2, the phrase is a feature of the language, specific to English, whereas figurative uses of the sort in stage 1 are more like instances of what Stephen Pinker and other linguists call the "time orientation metaphor," which is a feature of language generally, although the way in which it manifests can be different in different languages.
Stephen Pinker discusses "time orientation metaphors" in The Stuff of Thought: Language as a Window into Human Nature.
In the TIME ORIENTATION metaphor, an observer is located at the present, with the past behind him and the future in front, as in That's all behind us, We're looking ahead, and She has a great future in front of her. Then a metaphorical motion can be added to the scene in one of two ways. In the MOVING TIME metaphor, time is a parade that sweeps past a stationary observer. The time will come when typewriters are obsolete; The time for action has arrived; The deadline is approaching; The summer is flying by. But we also find a MOVING OBSERVER metaphor, in which the landscape of time is stationary and the observer proceeds through it: There's trouble down the road; We're coming up on Christmas; She left at nine o'clock; We passed the deadline; We're halfway through the semester.
Pinker goes on to point out that in Chinese, vertical metaphors where the future is "down" and the past is "up" are common. In other languages, the future is considered to be "behind" the observer, while the past is "ahead."
In early uses of the time orientation metaphor "down the road," the figurative context leaves room for it to refer to the future or the past. In this example, "down the road of time" means that something has been relegated to the past.
We have done our best to keep track with the acts and doings of the General Assembly, and we are under obligations to Hon. R. S. DesChamps for furnishing us with daily copies of the Journal and Calendars, otherwise we should have been dependent upon the reports in the daily newspapers which are not always correct: the Journal is the official guide.
The session is over, it has passed down the road of time, never to return, but while this machine was in view, it made a whole lot of noise.
An even earlier use refers to a "road of time," though not in the phrase "down the road."
There are wells on the great road of time, the prospect and certainty of which encourage the traveller to surmount the difficulties of his journey
Determining when the phrase was adopted widely enough in a figurative fashion to become idiomatic seems to me to be somewhat subjective and difficult to trace. I'm inclined to believe the dictionaries that attest its earliest print uses in the 1950s or '60s. From the OED:
It was a sad good-by for all of us. But one good thing, we know we'll always be seeing each other down the road.
- 1964 Mrs. L. B. Johnson White House Diary 17 Nov. (1970) 204
Some earlier uses appear in the 1950s:
Some 50-odd years later down the road the executive mansion is manned by Harry S. Truman.
No drastic action is required on the problem as of now, McElroy said, but it must be considered later, "down the road."
That the phrase "down the road" is placed in quotation marks in the second example seems relevant to the evolution from a general figurative use to a more idiomatic use. The quotation marks indicate both that the writer felt the need to attribute the quote to the speaker, suggesting that its idiomatic frequency had yet to fully develop, and that the phrase was used idiomatically in speech, which one would expect to take place before its prevalence in print.
Mixing Time and Space
There are some interesting uses of the phrase that might be considered a sort of partial-figurative use. Consider this quote where the phrase "later down the road" refers to a person's literal position in both space and time. As a person drives down the road, they move forward on the road literally, and forward through time in the time orientation metaphor. One might usually say "Further down the road," but since it will take time to get further down the road, saying "later down the road" can be interpreted literally to mean "at a later time, when they were further down the road."
Fifteen miles later down the road, Haynes decided to wake up Tatum.
Considering how uses like this blend literal senses of time and space, it makes sense that the time orientation metaphor is such a common figurative aspect of language.