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.