Well, the question itself is problematic. There is no one "American" BBQ. It's regional, and can be very different between regions in the USA. St. Louis BBQ is very different from Memphis BBQ or Carolina or Kansas City BBQ....
Take Texas, for example. You have your East Texas BBQ, which is very much like Southern BBQ (I seem to remember a sauce that was sweet), you have your Central Texas BBQ, which is heavily into smoke - you don't slather it in sauce to cook it, you smoke your beef brisket or ham for a couple of days, typically on Mesquite or Oak wood (or both). You will have a sauce to go with it, but that's added afters. There's South Texas BBQ, which, to me, has a heavy Mexican/border inflection.
Then there's West Texas BBQ. If you draw a line following the line on the East side of the panhandle down through the state, west of the line is West Texas. In the southern part you'll get Mesquite-smoked BBQ (again, cooked separately from the sauce) because Mesquite grows like weeds out there - literally. It's everywhere, and no other trees really grow out in that part of the state, and the Mesquite only gets to large bush stage (lack of water) usually. The beef will have smoke color in a good 1/4th to 1/2 inch, and the crispy bits on the outside... are to die for. Go farther north, through Lubbock and you start running into a slightly different set of spices, but still smoked with sauce on the side.
I don't know if they sell it in the UK, but if so... when I was younger and was in Lubbock for a while, Mr. Stubb (short for a longer name) had a little hole-in-the-wall place in East Lubbock that was the best in town. He long ago retired, but I recently saw that he sold his recipe and you can now get "Stubbs BBQ Sauce" in the grocers. With his picture on it and all. (I hope he made mint on that deal!) I'm going to have to try it to see how close it is to the original. Lubbock back then was segregated, so nipping over to the BBQ joint meant crossing into the wrong side of the tracks. Mr. Stubb was always so nice, and I always seemed to have more on my plate than other people did - I guess it was because I was poor and skinny in those days and he hated to see anyone go without. (Bless his soul.)
My experiences with Amarillo BBQ have been that it's been influenced by elsewhere - the few times I had it, it had been served ~in~ a sauce. (Sadly, not a particularly good one.) Out in far West Texas, by El Paso, and you again see the Mesquite, but it's just a bit different. Probably again due to the Mexican influence. (Not at all bad, but just a bit different.)
I have to say, the best BBQ I ever ate was from the Dairy Mart in Hillsboro (probably not there any longer, it's been several decades) in the Texas Hill Country. They had a big smoker out back and that thing was going night-and-day. Back before cars had air conditioning we could follow the scent in as we came in from West Texas with the windows down, like bombers following in a radio beam.
In all cases, served with a side of red beans and either slaw or potato salad. (Usually slaw.) And either Wonderbread or a white-flour roll. OH, heck. It's now late at night and I'm jonesing for some BBQ. There are very few places in San Diego (where I am now) that do a good BBQ. The one I'm thinking of is about 15 miles a way, uses Oak, but it's pretty darned good. And closed. (WAHHHHH!)
In any event, when someone tells you they're serving "American" BBQ, be sure to ask which one - or from which part of the country. If they look back at you blankly, it ain't "real" BBQ.
http://www.eater.com/2016/6/16/11889444/where-to-eat-barbecue-styles has an interesting take on it.
