I don't even know where to start. I have been tasked with migrating an older version of CiviCRM (3.4.8) on Joomla 1.5 to a newer version of CiviCRM on Wordpress. I got a fresh install of CiviCRM done on Wordpress 4.4, but I'm not sure where to go from here. I'm not familiar with Joomla (this is not my website), so please be very specific. I need everything (users, customizations, whatever else) moved over without having to delete anything on the current site.
For the question about Joomla (users and permission), ask the wordpress or joomla communities.
For the migration of civicrm itself, it should be fairly easy: just copy the civicrm database (or the tables civicrm_* if they are in the same database than the joomla site)
Check on civicrm.settings.php see if you have a mention of joomla and replace it by wordpress and you should be almost there
You might have to manually change how the drupal users are mapped to the civicrm contacts, unless you do manage to keep the same ids for joomla and civi
Re: civicrm.settings.php mentioned above, since you installed from WP, you shouldn't need to worry about Joomla there. What version of CiviCRM did you install in WP, the latest? If so, the schema has changed a good bit from 3.4.8! It might be best if you can update/upgrade CiviCRM to * (whatever version you are asked to use) in (a copy of) the Joomla site, and only afterwards, move your CiviCRM database into the WP- civicrm one. The upgrade can take some time, and present its own challenge. There's a table in CiviCRM syncing WP defined users and CiviCRM users, civicrm_uf_match. The users should exist as 'contacts' in civicrm, and the email address is the uf_name, or identifier, used to match users in WP and CiviCRM. If you have already created users or data in the 'new' CiviCRM database with WP, save that database and, after you do the upgrade, try changing the connection info/credentials in the WP install to point to it. Go to 'CiviCRM' from WP (it looks like another plugin) and the user data should automatically get synced, you can verify by looking at the uf_match table.