Instructions
Visit EU Battle.net. Click "Log In," located in the top right corner of the page. A new pop-up window appears requesting the email and password for your Battle.net account. Click the "Log-in" button. The European version of "World of Warcraft," along with any other Blizzard games licensed to your account, appears.
Click "Explore" in the upper right corner of the page. A drop-down menu appears. Select "Buy Games." All games that are available for digital purchase will appear in the middle of the screen.
Click the "Change" game region link, located in middle right portion of the page, directly above the "StarCraft: Anthology" box cover. If you have a European Battle.net account, it is set to European by default. Select "North America" from the region list. Click "Continue." You will return to the game selection screen for the North American region.
Click the box for the "World of Warcraft: Battle Chest." Click "Buy Digital Version." The purchase screen appears, indicating the cost of the game. Press "Continue."
Input your payment information on the following screen. Click the checkbox indicating "I have read and I accept the following Terms of Sale," then click "Continue." After your payment has processed, the North American version of the game will appear for download and installation to your computer.
Install the North American version in a different location on your computer than your European version to avoid software conflicts. Create a separate shortcut to launch the game on your desktop.
Launch the game. Select a North American server and create your character. You now have two installs that allow you to play or change back and forth between European and North American servers, provided you have a current subscription on both accounts.