‘On time’ is probably a more appropriate description since it refers to a very specific starting time. But I personally wouldn’t use either on time or in time in the sentence. I think on time was given mainly as a choice of definition, not as a typical usage in the sentence.
Thank you very much for your comments and directions, as always. I was like trying to say 'as they didn’t hit the road in time, they… I felt weird to say 'as they were not ready on time, they…