I've done ER work as a counselor for rape victims through a crisis center that had agreements with all the local hospitals so they didn't have to hire staff for it, and I think it probably depends on the hospital if that happens or they have on-site social worker or rape nurse or no one at all. I'd be called the instant they got her, get out of bed, rush down. She'd only be partway through treatment (for the important stuff--evidence isn't the top priority of medical people, her health is) and I'd arrive and stay, with her permission, through the rape kit. As juniper said, there is a variety of responses, but mostly they are 1) shocked (terrible to know, but the worst of the psychological effects often come at about 1 year after the event when most victims think they're going nuts, have many nightmares, what they see as irrational fears); 2) happy to be alive; and 3) worried about a loved one's response. (My mother told me this would happen/my husband will kill him and I'll lose him/etc.)
I had a victim who was stabbed 17 times in the belly (somehow, he missed every single important thing) and the least of the concerns was the rape kit. It got done in surgery; she was totally out for it and never specifically consented to it. I had a retarded epileptic victim who seized every fifteen minutes. The medical staff was busy with a major car wreck and they said, "call if she isn't breathing," and left me there to watch seizure after seizure. But that wasn't important enough for them to stick around and deal with, considering their other patients.
Back then, if the victim was white, a homeowner with a good address, the police would be there fast. They'd be there if they brought her. Everyone else--teens, the poor, black women, the police took their sweet time and the victim was often discharged or it was even the next day. Maybe that's changed, but if 911 wasn't called, there's a chance the police won't be there for hours.