Most, if not all, Infinity installs are self-install now, an engineer is not required. The hub is just plug and play, the engineer just re-arranges your connection in the green cabinet and the broadband is activated remotely.
Moved to correct board.
Until the phone is working, broadband cannot be provided.Have you receive notification that your phone line has been activated?
If so, there is either a line fault, or a problem within your own internal wiring. You can prove your wiring by plugging a phone in the test socket.
Ports are provided in the exchange, but its unusual to run out of them, as there are normally plenty of spares, as people move over to BT Infinity.
I would suspect a line fault, which is not unusual, but you must check your own end first.