Reproducibility was talked about by Robert Boyle in 17th century. Literate Programming was invented in 1984 by Donald Knuth. Reproducible Research tools like R markdown were invented recently. My query is : After the invention of Literate Programming, how did the trend of making reproducible reports start? How did tools from R Markdown evolve AFTER the invention of tools like WEB and noweb.
[This answer was written for the original question, which has since been substantially edited.]
The question is backwards: if anything, it's literate programming which would have been inspired by reproducible research.
The concept of reproducibility is closely related to the criterion of falsifiability. This concept and its significance in science obviously predates literate programming (and anything related to computer science). More generally, the scientific method has been a pillar of scientific research for a long time. In particular the importance of reproducibility in research is definitely not limited to computer science.
It happens that literate programming can be useful to address the specific problems of reproducibility in modern computer science research, especially in experiments involving machine learning and massive amounts of data (related example). But as far as I know it was not designed in this specific goal, even if it shares some of the principles of the scientific method (transparency, interpretability).