A lot of people have asked, "which state are you from?" Or "where are you from in the States ?" Like they know already. I have a pretty southern accent so I guess it isn't confused with Canadian or something like that. But if people ask me where I'm from I've been saying, "I'm from Louisiana but j live in Swindon now"
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I think when I first moved, people almost always assumed I was from the US, but as my accent (Boston, might I add, so not necessarily pleasing to the ears for some LOL) died down, I got questions about where I was from and I'd say "originally the US" and they'd respond with "Oh you had a bit of a twang so I wasn't sure if you were Canadian". I even had people - off the back of that - ask if I was offended as an American being asked if I was Canadian......I'd reply with "not at all, it's similar enough in many ways I s'pose" (plus, it takes a lot to offend me in general) and I genuinely had responses of "Oh, because I know a Canadian would be offended if you asked them if they were American"..........Uh....oh...kayy......? What do you say to that?!
LOL now I can mostly fly under the radar. I guess I sound like I'm from Bristol/West Country - or so I'm told. I still get asked where I'm from but when I say "originally? From the US." I get a "Oh! some words you did have a twang which is why I asked, but I thought West Country/Bristol! You don't sound American!". I then have some people telling me I still don't sound English while others comment on how English sounding I've become since they first met me. I don't even know anymore hah
I reply more or less like you if I want the Convo brief. I always assume the person doesn't really care about background (or at least not the whole story) unless they keep asking.
Here in England I say I'm from the US. Most of the time they ask where from the US. I say Florida. Convo finished.
When I go on holiday I still say I'm from the US but live in England. I say this because I've had people tell me I don't sound British. You don't say... 
At work people tend to dig deeper as they can hear I sound American but still have a slight Spanish accent. That's because I was born and grew up in Puerto Rico (speaking Spanish) and didn't move to Florida until college/uni.
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Yeah if I can tell they don't really want my life story and it's just a quick question in passing (like at work), I'll say " From the US originally" (because I am told I now have an accent, the conversation rarely stops there). If they press to ask where in the US, I will typically say Boston over Massachusetts because they are more able to place Boston than the state as a whole, oddly enough.
When abroad, if I want to keep it brief and it's my husband and I being asked, I say England or UK. If I'm in England and somebody asks in passing and I can't be bothered I will tell them the name of the village I live in.
At most, it's just as I said in my original post of "Originally from the US but live in England/UK/here now" and avoid going further into detail unless pressed (which can sometimes happen now because they are curious why I don't sound like a 'typical american'.)