How do I update an unlocked phone without the SIM card?

If you've requested an unlock from your carrier, but have not got the original SIM card, you will need to reset the phone before it will be unlocked. First ensure that you have requested an unlock from the carrier. Now follow these steps

1. Back up your iPhone. Either using Backup (Settings > iCloud > Storage & Backup > Back Up Now) or by connecting the iPhone to a computer and Using iTunes to perform a backup.
2. Erase your iPhone. Click on Settings > General > Reset > Erase All Content and Settings.
3. Complete the setup assistant and restore from your backup.

The iPhone should now be unlocked.

When you turn the iPhone back on, complete the setup assistant and restore your iPhone from the backup you created. Then you can insert the new SIM card and it should now work. So there you have it, your iPhone is unlocked