Do I qualify for a UK passport\citizenship?
See Apply for citizenship if you have a British parent/You were born before 1983. Since you did not mention your father, I presume he was not British. This would mean that you are not automatically a British citizen, but you can apply:
If you’re not automatically a citizen
You may be eligible to apply for citizenship if either:
- your parents were not married when you were born
- your mother was British, not your father
The second is true, so you may be eligible.
Your mother or father must have been a citizen of the United Kingdom and Colonies when you were born. They must also have been one of the following:
- born or adopted in the UK
- given citizenship after applying for it in their own right (not based on having a British parent)
- working as a Crown servant when you were born (for example in the diplomatic service, overseas civil service or armed forces)
The first is true, so you meet this requirement as well.
You must also be of ‘good character’ - read the guidance when you fill in the application form.
There is a link to apply online if your mother was British.
Even if I do, would I be able to live anywhere in the EU, since the UK is part of the EU?
Yes, under certain conditions. To stay longer than three months, the host country can require you to register, and they can require you to be in one of the following categories:
- working (including self employed)
This is laid out in directive 2004/38/EC, the freedom of movement directive. There are many questions on this site about that.
What about Brexit?
After Brexit, nobody knows, but there will probably be a system to allow British citizens living in the EU at that point to remain without too much burden. It seems likely, however, that British citizens newly moving to the EU after Brexit would face a situation substantially similar to that faced by US citizens moving to the EU.