Both. And neither.
I moved to the Greater London / Essex area in Spring 2006 at the age of 24. I had what I think was a terrible start with trouble finding a job, a GP who wouldn't listen to me regarding my depression, and a lack of any social circle. It's a fairly normal start by UK-Yankee standards I believe
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My husband is an only child and both of his parents are very hands-off so settling was hard. I gained citizenship after 3 years and finally started to feel settled and content after about 4-5 years. It was about 7 years before I really hit my stride and thought, this is where I belong.
I go "back home" to Georgia the same way my mother "goes home" to Ohio despite leaving there in the mid-1970s. The UK is my forever living space though. I don't see leaving and moving to the US.
The UK isn't really home either, at least not in the completely understanding it sense. I sometimes just don't get things. I still get frustrated and baffled by things, but less as time goes on. I'm occasionally reminded that I have two homes where my heart is and that can make me feel like I don't fully belong anywhere in my greatest moments of frustration when I feel I'll always be an outsider in some way.
In a less emotional sense, home to anyone else is simply in context. I bring shopping home from the grocery store (that's to London!) just as easily as I say I won't be coming or going home in the Spring due to work after all (that's to the US!). I won't let anyone else host Thanksgiving because I need it to be like it is at "home" (US) but I also won't eat out then because it needs to be at "home" (my house). Both my parents and my mother and law use home quite fluidly having moved far away from their childhood location and all their siblings. There's rarely any confusion.