There are some branch names which retain elements of their provenance - for example Nottingham and Lincoln both have a branch of Natwest named ‘Smiths Bank’, which is a remnant of their original purpose as branches of Samuel Smith & Co.
Just a cute name tho, totally standard Natwest otherwise.
There was a historic Barclays branch on Fleet Street too, Goslings, but the branch closed in 2017 and as far as I remember despite the cool signage it was just a branch of Barclays, with Barclays branded stationary etc.