I am one of those Americans you seem to be so horrified by. However, let me answer you anyway, as a fellow grandmother (and great-grandmother!)
My family (3 children, 6 grown grandchildren, one great-grandchild) all live on the same continent as I do, but that's all I can say about our physical closeness. We are scattered all over the country. One son and his (grown) son live sort of near me, that's it. And young people seem to be in a frequent state of flux, moving from one location to another much more frequently than previous generations did.
But we still remain a very close family. We are constantly in touch via texting, phone calls, Facetime or Skype, email, pictures (I have literally hundreds of pictures of my adorable great-granddaughter), and frequent visits whenever possible. Throughout the pandemic, we had weekly Zoom meetings. A few weeks ago, we all shared a rented house at the beach for a week; everybody came, including bringing significant others, and even a few children of the significant others. All were welcome.
In this day and age, moving from England to America is not that different from moving to the next town or county. It's become a very small world. You can still stay in touch, speak every day if you choose to, have discussions, share jokes, ask for advice, etc. All the things you would do in person, except physically hugging each other. And you can do that when you actually visit each other.
So lighten up, Granny. If she moves to America, you can still be in touch. Any problems in this situation will be caused by you, if you continue with your current attitude.
And good grief, it hasn't even happened yet, it may not happen, and you're already in a tizzy about it!
P.S. In spite of what the news media are focused on, we are not all shooting each other in the streets. The gun laws are much too permissive, but all in all, life is usually pretty safe.
Think of the glass as being half full.