I've just looked at bus timetables
and looks like there's a 55 leaves Keswick at 8:39am
to Ambleside for 9:15
then an X33 from Ambleside at 9:26am which goes through Coniston to Broughton In Furness (and onwards to Ravenglass), which reaches a stop called "Road To Woodland" (which looks like a short walk to Broughton Mills) at 10:14am
this looks fairly doable
am I missing something?