One possibility is in Cumbria, but north of the Lake District. Carlisle is an easy train ride south from Glasgow, and while the city of Carlisle itself may not promise dreariness, it has the truly outstanding Tullie House Museum, with fascinating artefacts from when the area was part of Roman Britain nearly 2000 years ago. A temporary exhibit, going on right now, includes 2 magnificent Roman cavalry mask/helmets, which cavalry officers would have worn to show off while on their horses. There's an impressive castle in Carlise, too, and a medieval cathedral with surrounding priory land and buildings. Rain and/or overcast skies are significant possibilities.
The Hadrian's Wall Path national trail runs through Carlisle, and you could head west (towards the sea, even crossing an area that floods at certain times of the year!), or east into more farmland, where you can ponder the sheep and cattle as you cross their fields. Close gates as you pass through to the next field, and there are plenty of places to pull up a seat and contemplate the hills in the distance and the surrounding rivers and countryside. It may be a bit of a challenge getting out of town on the trail, though, as riverbank landslides have caused trail diversions, and the signage for where the trail's going now is spotty. A good trail guidebook and asking locals (and maybe calling a taxi if need be) can keep you on track. If you venture far enough to the east, you may even glimpse some remnants of Hadrian's Wall itself, and parts of its forts, milecastles, and turrets. Enterprising farmers and church-builders over the centuries tore down much of the wall to use stones for building their homes, barns, and churches.
Many towns along the Path have serviceable pubs, although not every one is open every day, and early is better than late, as many close by 9PM. The people in the region are fantastic, as nice (or nicer) as you'll find anyplace!
