My wife is a midwife assistant.
The Amish do go to the ER when necessary. In our area "necessary" is often left to the decision of the midwife. Some Amish preferred having a birth in the hospital, covid has dramatically changed that though and numerous birthing centers have sprung up to accomodate that.
The cord is cut at the time of the birth and clamped. The Amish men usually want to cut the cord and take care of the placenta, which usually means that it gets buried. There are some that will request that it be encapsulated and brought back to them. I don't know the details of the reason for that, maybe there's someone on here that can explain. There might be some ultra conservative groups that have no midwife input, that out of ignorance they just let the cord fall off, but it is certainly becoming less and less as there has been significant effort on all the Amish groups to teach safe birthing practice.