The first place to start is to work out if you know your nan's birth name. That is the name that she was registered as when she was born.
If you do, then that makes things very simple. All you need to do is to go to the General Register Office website and you can get a copy of your nan's birth certificate which will show her mother and also her father if they were married.
If you don't know her birth name, and you only know her adopted name then things become a bit trickier. The General Register Office holds the information that links adopted names to birth names, but they don't give this out to the public.
Instead, you have to use an "intermediary agency" and pay them for doing it. It is explained here in detail:
http://www.adoptionsearchreunion.org.uk/contact/intermediaries/questionsanswersrelatives.htm
This should give you details of who the birth parents were and also possibly information about why your nan was adopted.
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With regard to your original question about Ancestry DNA tests here is a map that shows all the different regions that they differentiate:
https://support.ancestry.co.uk/s/ancestrydna-regions
Your DNA results will come from a number of these different regions.
One thing I would suggest if you do decide to do a DNA test is to also get your mum to do one as well (I presume that your nan has passed away?). The reason for doing this is that since she is a generation older than you she is a generation closer to your nan and so shares more of her DNA than you do.
This also means that you are likely to find more relatives of your nan from your mum's DNA than from your's.
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For example, on Ancestry it says that I have 21,700 DNA matches.
Both of my parents have done their DNA on Ancestry as well and my dad has 20,900 matches and my mum has 24,700.
So, by going back to an earlier generation it has more than doubled the number of DNA matches. From 21,700 for just myself to 45,600 from both of my parents DNA.