Her accent is so bad that it's distracting.

Where are you from? I'm English and have met people who sound like that...though, admittedly, a little faster

So, I just write something here and it will appear on everything I post?

As someone from London, I can attest there's nothing wrong with her accent at all, especially seeing as how it is actually her natural accent - her american accent was adopted as a child when her family moved back to the States. Perhaps you're just distracted by her tone, how slowly she speaks and the low register of her voice, but accent-wise, she's just fine

Yet another Yankee commenting on accents in the UK. LOL

I've noticed this a lot. People in the US have an idea in their mind of how people from a certain country are supposed to sound and if someone doesn't sound that way they assume it's bad acting. For instance, a lot of people think all Aussies are supposed to sound like Crocodile Dundee.

Whenever there's a show or movie where an American or Canadian actor plays British, you only have to wait a short while for the inevitable thread saying their accent is bad, always by an actual British person.

It's just something that is not commented on nearly as much when a British actor plays one of us. Only when extremely badly done.

Gillian lived in the UK for so long that it had to be more authentic than someone just faking it.

 That's right! You're about to be killed by a zamboni! ⛄

Her accent itself is okay. I think what you're picking up on is her speech cadence. It's a bit too deliberate and you may be subconsciously detecting that she's acting, especially in comparison to all the other actors on the show who "feel" more organic in their presentation since it comes more naturally to them. This may be why it feels off to you. It's not necessarily about her accent, but her speech pattern. At times it comes off as a bit unnatural for me as well, but it's not very distracting for me. It's possible that this may be more apparent to, or more acutely perceived by, those who are accustomed to hearing her talk in her natural manner, which sets up expectations in the brain that are then defied when hearing her talk with this different accent on "The Fall".
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I'm something new entirely. With my own set of rules. I'm Dexter. Boo.

Her accent is 100 percent authentic. My Auntie sounds just like her. Sarah Ferguson (Fergie) has a similar accent. It's the sort of accent you mainly hear in South East England, typically public school educated (known as private school educated in other countries), from a well-to-do educated family.