I LOVE The Eagle and Childe!
@SouthernHope My husband and I did a mini version of what you are planning, carrying our then-baby strapped in a baby backpack. We were in the Peak District, which I recommend highly. It feels a bit misty and wild with some very authentic untouristy villages, though Chatsworth House is there, which is the mother of stately homes. I also absolutely love the Lake District. I’d say the Peak District is a bit less commercial and feels more rural. The Lake District, IMO, probably has more to do in terms of tourist sights and there will be plenty of traffic. However, there is most definitely real, strenuous hiking in the Lake District and if you go in the height of summer, there will be a traffic jam of hikers on the most slippery and shale-y parts of the trail.
I also recommend the Yorkshire Dales (Bronte sisters country) and Exmoor and Dartmoor, which feel remote and wild but really aren’t far from anywhere. For a very different kind of pub crawl experience, go to Cornwall, which will have an olde seaside and pirate-y feel. There’s a fantastic coastal walk that will take you to Land’s End, where there is, unfortunately, a ridiculous “tourist village” with paid entry. But the coast all around there is spectacular and there some truly breatkting and magical things to see. Mousehole (pronounced Mowzle), St. Ives, Mt. St. Michaels (you can only walk out to it at low tide,) Penzance, Marazion and Sennen are all worth stopping at. There will be pubs scattered all around the area, though I suppose you will want the most charming and olde worlde ones.
Anyway, I haven’t exactly answered your question, but I hope you might at least get a better sense of the English countryside and where you might want to explore. Sounds like a fun trip!