Cornwall and Devon or the Lake District

Might help if you said why you chose those two areas and what you expect to see.

They are both nice; if you've no strong preference toss a coin.

We plan a four week tour that includes South Wales (ancestors from there) as well as much of England, except London (done that) and the Southeast. We are planning on York, Durham and Hadrian's wall in the NE and could swing by the NW and lake district, but Cornwall is appealing as well.

Cornwall and Devon are diagonally as far away as you can get from where you will be in the northeast. The Lake District is nearly next to where you will be.

If you have no specific reason to go to either go to the one next door.

Since you're already going to be in the Northeast, I agree the Lake District makes more sense than Devon/Cornwall. You could follow Hadrian's Wall from east to west as far as Carlisle and then turn south to the Lake District. If you're interested in stone circles, there's Long Meg & Her Daughters outside of Penrith, and Castlerigg near Keswick. Keswick as a town is much more touristy than Penrith -- too touristy for my taste. I enjoyed Penrith better; recommend Narrow Bar as a wonderful little café for lunch.

You mentioned you're also going to be in south Wales, but even though that is relatively near Devon/Cornwall as the crow flies, the land transportation would not be very direct. Personally I'd put Devon/Cornwall on a separate itinerary with Bath, Glastonbury, and the Salisbury Plain.

We plan to visit Bath and Gladstonbery on our way to Wales. We also plan several days in the Cotswolds then on to Nottingham, York and points north.

We could easily go Cornwall before Bath, Cardiff and South Wales.

Personally, I'd do Cornwall over the Lake District. Been to both and prefer Cornwall. The Lakes, not as much.

I would try for both over a four week period.
Week 1. Durham. York. Lake District.
Week 2. Wales.
Week 3. Cotswolds and Bath.
Week 4. Devon and Cornwall.