Is there a right way to go to an emergency department of a hospital? Are you not supposed to turn up on your own?

Two years ago today, we were staying in a hotel, we were back there between a wedding and reception when DD suddenly got very sick - lots of vomiting, very hot. We called the nurse on call number but he wasn't at all helpful, so we asked if we should just go to hospital. He said he could call an ambulance if we wanted. We declined, called the hospital and drove there. It wasn't the kind of emergency that warranted am ambulance, but they seemed very confused by us driving ourselves there!

Then on Friday, DS got sick. I was at the GP with him and he said he needed to be in hospital. We went round to the nurses room and he was put on oxygen. I had said ok, I'll take him straight there, but they didn't want him in the local one so we took an ambulance to the big one. I didn't question it.

But the way they reacted when I said I'd go there, is it not normal to take yourself there?