Most people think it takes around 2 months to transfer a username. I'll tell you how to do it in two days (rough estimate). This is pretty much commonsense.
Rules:
Both accounts must be P2P.
You must have forum access.
First off, turn your private chat to 'off' as well as your friends list. Hide on RuneScape, so it lowers the chances of your name being seen or noticed.
Change the name on the account that contains the username you want to something offensive e.g 'BigDickPussy'. When your name is offensive, JaGex will reset your username to numbers, thus erasing your previous name before the offensive one (The (rare) name). Post in forum sections such as Jagex Lite, General and other busy sections. Posting on active stickies where Jmods can see your account is essential to a fast reset. You will most likely get temp-muted from the forum. That is a good sign. You shouldn't recieve any blackmarks for this.
It will take around 24hours minimum for your name to reset. You must be active, and watch this carefully. Once your name has been reset, the other account will be able to access that username you wanted to transfer.
Also, DO NOT TELL ANYONE that you are changing your username. Don't trust anyone not to steal your username. I've had names stolen in the past from untrustworthy friends.