We're thinking of traveling to Scotland and Northumberland in September. We'd land in Glagow, take the train to Edinburgh, spend a few days there, then take the train to Durham to see the castle and cathedral. All of that should be easy.
We'd then like to see Alnwick castle and gardens. We'd take the train to Alnmouth, take a taxi to a hotel, and walk to the castle and gardens.
The next day we'd see the Cragside NT property and perhaps Rothbury. We'd travel by taxi there and afterwards back to Alnwick, pick up our luggage from the hotel, taxi back to the train in Alnmouth and head north for more time in Scotland.
The main question is whether there are adequate taxi services in the area to make this work efficiently. If so, please recommend some companies or, if applicable, apps. We do not want to drive or take a bus. Cost is not a major issue.