My DH is a paramedic, and if he can, he will always take the patient to the mat hospital they are booked into. Unless he thinks birth is imminent or there are other medical reasons to go to a different hospital.
He also said that you don't triage through ED, he would take you straight up to maternity.
The only time you would go in to ED is if you were just say - crowning on arrival, meaning that they would get a midwife to come down to assist you to deliver,

It is basically the same ad if a friend or family drove you in.

He also has on occasions delivered in the back of the ambulance. If that happens, they pull over and both medics assist eith delivery.
Hope that helps.