I can only tell you what my preference would be. There is no right or wrong answer here, only different choices. If you ask ten different Forum posters this question, you will get ten different answers!
My preference would be:
Day 1--Leaving Edinburgh, go to Bamburgh Castle. An impressive medieval castle overlooking the beach and sea. Go from there to York, a walled medieval city. You can walk ontop the ancient city walls here. The city was occupied by the Romans, the Vikings, and York has museums to explain their long and fascinating history. Walk through the streets of The Shambles, a section of the city unchanged since medieval times. See York Minster, one of the greatest cathedrals in England. You will want to stay in York the night you arrive here, sightsee the next day, and possible stay a second night after your long day of sightseeing. So my preference would be to make a hotel reservation for two nights.
Day 2--Leave York early, head south down the M1 roadway. When you get closer to Leicester, take the M69 motorway toward Coventry. Go to Warwick and park in the Warwick Castle parking lot. Warwick Castle is an interesting medieval castle. Walk around the town of Warwick; lots of interesting buildings.
Continue on to Stratford-Upon-Avon for the night. Stay here one night.
Day 3--Walk around Stratford-Upon-Avon, see Shakespeare buildings. It is on the River Avon which is connected to a canal. Lots of canal boats here. Take a boat ride. Drive from here south into the Cotswolds and spend the night.
Day 4--Drive around the Cotswolds. You could spend two nights in the Cotswolds.
Option: Drive to Oxford and spend one night there. Walk around Oxford and take in the sights.
Day 5--Drive from Oxford (or the Cotswolds, if you have chosen to stay there longer and have skipped Oxford)
I think your teens will like the castles, York and its Viking history and walking York's walls.
If you choose to stay two nights in the Cotswolds, there are plenty of villages to explore and walking trails.
One village that may be of interest to your teens is Bourton-On-The-Water. It has a small river running through its center, plus good cafes, a shop to get ice cream, and a good bakery.