As a UK national, you are entitled to give birth here under NHS treatment. No one will 'flip out' for that.
Why don't you talk to your family about getting registered w/their nearest surgery? That would be a good place to start. Your local surgery will usually have community midwives attached to it who will treat you during pregnancy and birth.
Where you give birth - which hospital - depends on your location. In mine, Edinburgh, for example, there is only hospital w/i the city limits in which to give birth: Edinburgh Royal Infirmary. But you can also opt to go to a nearby town, Livingston, and give birth there.
As a UK passport holder, you have no immigration issues. You simply enter the UK on your UK passport.
As your child will be born in Britain to a British mother, he/she will also be a UK national. Being born on UK soil does NOT guarantee a person UK nationality. Your child will derive his/her nationality from having a British mother.
Before returning to the US, you will need to get him/her a UK passport, as a British person must enter and leave the UK on their British passport.