If you bought it from a carrier phone, it will be locked to that carrier. It will lock to the first SIM temporarily, until you connect it to iTunes and it gets the new lock status. This is because the lock isn't applied until the phone is registered on the sale, and this is often done in batches on the backend, so until this has been done, your phone doesn't know what it is and defaults to "first sim" mode.
If you buy from anywhere other than Apple or a carrier, it will always be "locked to first sim" mode.
EE's unlocking page is here.
http://ee.co.uk/help/getting-started/joining-ee/unlocking-your-phone
It depends on your service type (PAYG is £20), contract they will only unlock the original handset.