I can transfer money between my Natwest accoutns online and it shows up same day.
I can pay (for free) money to other bank accounts and also to pay bills.
You could possibly phone your bank and tell them to do it that way - or go into the branch, withdraw the money and pay it into the other bank branch - in that case you would only be walking around town
Don't always expect them to count the cash in private tho - they once counted 2k out on a normal counter in front of everyone!