I had this happen myself.

DD3 came unplanned at home. Well, not entirely unplanned; the planning happened about 1/2hr before she arrived as I flat out refused to go to the hospital.

My biggest fear was the placenta getting stuck or excessive bleeding. Once Lucy was out, DH called the ambulance to come. I kinda wish we didn't call because it really was pointless. The placenta came out perfectly intact and there was no heavy bleeding. I really could have popped bub in her carseat whilst I sat next to her holding the bowl that had the placenta in it while DH drove us to the hospital to get looked over.
The paramedics had nothing to do but clamp the cord and give DH some scissors to cut it. They then wrapped Lucy whilst I had a shower. They where here for a while as I was in no rush to get her off me.

The only other thing I wish we were more prepared for was taking photos. I don't know how we could have done it as DH caught Lucy, but it would have been awesome to have some 'being born' pics like we have for the others.