Originally Posted by
stifle
...Most banks will not want anything to do with you if you are not a customer, and it's unlikely you will be able to exchange Scottish for English notes.
I may have just been lucky in my choice of banks, but I have never had a problem exchanging them at whatever bank I walked into in London. I have never been asked if I had an account there; from my U.S. accent, it would be unlikely. Granted, this is entirely anecdotal, is based on a small sample size, and my last visit to Scotland was in late 2014* so things may have changed since.
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*Up to and including September 18th. It was interesting, to put it mildly.