Comment by prennert

You probably found it already when doing your background search, but there is an acoup blog post about why the Industrial Revolution did not happen during the Roman period: https://acoup.blog/2022/08/26/collections-why-no-roman-indus...

There might be some inspiration in there to guide the story towards breaking some of the chicken and egg problems. Maybe the Romans find a way (and reason) to exploit the English coal deposits and start encountering the same problems the English did eventually: how to pump water out of shafts.