That sentence you presented still won’t make sense even if the words “years old” are added to it, as follows:
“For those who were 18 at the time the accident happened, the three years does not start until they turn 18 years old.”
The above sentence is illogical. If those referred to were already 18 years old when the accident happened, all of them had already turned 18. The main clause, “the three years does not start until they turn 18 years old,” is therefore grammatically and semantically faulty. The verbs “does not start” and “turn” are in the wrong tense, making the timeline for the whole sentence wrong.
That sentence would make sense if the tense for that main clause is changed as follows:
“For those who were already 18 at the time the accident happened, the three years would start from the day they turned 18.”
“Would start” is, of course, in the past conditional, and “turned” is in the past tense. In this sentence construction, the sense is clear even if the phrase “years old” isn’t added.
When making statements of the kind you presented, it’s important to keep in mind that in reckoning with the age of a living person, that age remains the same for a whole year until that person’s next birth day. For instance, the age of someone born on March 6, 1995 will remain 18 from March 6, 2013 until March 5, 2014. The day after, March 6, 2014, that person will turn 19. In short, a living person’s age stays the same for one whole year.