The 'Tynedale' referred to in Hextilda's family isn't the Tyne in East Lothian its the Tyne in Northumberland.
Tynedale (down which the South Tyne flows) runs from Haltwhistle to Hexham. The giveaway is in her saxon name. She was named after the town she came from. She also donated lands to the monks at Hexham Abbey.
At the time, Tynedale was the border between Scotland and England.