Tennessee is beautiful. If you do go, make sure to visit the Great Smoky Mountains. My concerns would be a bit different than most British, although I'm also an immigrant to the USA. If you move here, you need to keep rabies in the back of your mind. Not something to panic about, but people on mumsnet seem to be very relaxed about handling stray cats and so on. You cannot do that in the USA. It's not a big deal (rabies infections in humans are very rare) but something to be aware of. If you are bitten or scratched by an animal you don't know, or wake up with a bat flying around the room, you need to get rabies vaccines.
You may need to use insect repellant if you spend much time outside during certain parts of the year, because of lyme disease and mosquito borne diseases like West Nile virus. You also need to learn to identify poison ivy and I guess also poison oak. You need to learn how to be bear safe and to identify any local snakes and spiders, especially when hiking. Again, this doesn't mean panicking, but just adjusting how you do things and being aware of your surroundings.
Also be prepared for extreme weather. Extreme heat in the summer, very intense thunderstorms at certain times of year, and the odd tornado and so on. Fortunately, with cell phones, you can get weather radar in advance, so you can modify your plans based on that, set your phone to get emergency alerts for thunderstorms and tornadoes and so on. Keep a supply of emergency water in the car and switch it out periodically.
Unlike some others say here, yes, you do have to worry about guns, but not so much about being shot at the mall. But if you go hiking, many hiking areas allow game hunting with guns, so that is something you have to pay attention to. Some areas prohibit hunting on Sundays so see if your area does that; if it does, then Sundays should be safe to hike. (The deference to hunters here drives me nuts. What about the rest of us who want to hike in peace? Although some hunters are subsistence hunters, most are just hunting for sport)
I'm making it sound terrifying but actually it's fantastic. I think it would be a great experience, for a few years. There are many beautiful places in the USA and lots of fun things to do. But it's not a tame country, even in places that try to look like quaint English towns. Deep down, the USA is a very wild place.