From the perspective of an Englishman...
I can't provide citations for this, but over the years there has been more ... pressure might be too strong a word, but encouragement from all sides, government, media, that we are all now British not English. It's what you put on forms, official or otherwise, it's what is offered in drop-menus for anything that needs a nationality.
English isn't there as an option [neither is Welsh or Scottish].
When I was asked in the US [oh, yes, it happens a lot] if I was Australian [how they can confuse the two will elude me forever] I would answer that I'm English...
... but that was 20 years ago, since I was last in the States.
I do get the feeling that British would be first to hit my lips these days.
There is the additional point that the actress's British accent is very much a modern Southern/London 'street/youth' accent and the glottal stop in "Bri ' ain" just hit the spot sonically better than "England" would have done.
In the UK, there is a great awareness of accents and the geographical distance between them. As a Northerner, if I'm speaking to another Northerner, I'll be specific about Northern accents and their respective towns/areas. If I'm speaking to a Southerner [my partner, for instance] you accept that they really can't tell the difference to such small degrees, so you broaden these areas. Living with this large set of accent distinctions in what is, compared to the US, a tiny area gives the average Brit a good sense of 'when to be specific and when to generalise'. We probably do it sub-consciously.
Regarding comments already posted - there is an additional consideration.
The question was specifically "Are you from [country]?".
The answer to that question is "No, I am from [country]".
The character is replying directly to the form of the question. She also has sufficient command of specific regional English to be able to achieve a very very convincing accent; she would be unlikely to slip up.
Were we still in the days of the Raj, then the answer would probably be [imagine stiff upper lip] "By god, no, I'm English!" with an implication that for anyone to assume otherwise is just because they don't know any better, being foreign.
That mental attitude [fortunately] doesn't survive to this modern day, and may do even less in the future...
...and bear in mind that all Black Mirror stories take place at some indeterminate time in the future.