If you are connected through Facebook, you have to log out in the game's Settings - User - Log out. Then restart your game, and AFTER THAT you can delete the game from your phone, install it again, and re-connect to FB. Only if you do it this way, the game won't offer you to log in to your old account.
Google Play
Same rules apply as with Facebook.
iCloud
If you are connected through iCloud, the data is stored locally, so all you need to do is to go to your iCloud settings and delete the WarFriends folder.
Game Center
To start anew, you have to go to your phone's Settings - Game Center - and log out of your Apple ID and create a new one. Then you can start a new game in WarFriends.
Please be aware that we cannot help you with disconnecting your existing accounts. The most secure solution is to create a new Facebook/Google Play/Game Center account and start your new game there, that way you’ll have a backup for all your accounts.