I am also from the US and came over to London to get my Master's Degree in 2006 so I'll give you some advice from my experience. I completed my degreee and now live and work in London.
Your starting point should be the British Council: http://www.educationuk.org/pls/hot_bc/page_pls_all_homepage
You will find a wealth of advice on how to pay for school, how to sort out your student visa and other important needs.
You will need to contact the school you would like to attend and get all the information from them - they will have a lot of information that you will need, as well as some that will be helpful to you throughout this process.
You can get loans that are good for a university outside the US, I applied for the FAFSA (Free Application for Federal Studen Aid) as well as taking out private loans through my bank to finance my housing and living expenses.
Even though the exchange rate is a bit better than it was when I moved here, it's very expensive to live in the UK. Glasgow may a bit cheaper in terms of housing, but the general living costs are similar. Make sure you do your homework to fully understand how much money you need while you are here.
Students are not permitted to take full time or permanent employment - so even if you try to work, you will most likely be unable to make enough to support yourself.
You should contact the school you are looking to attend and find out their estimated costs per year for both tuition and living expensve. Most banks will require that information from the school before approving your loans.
An American citizen cannot simply choose to apply for dual citizenship. Becoming a British citizen is a long, complicated and often expensive process. Unless you are born in the UK or have a British mother (even that is changing now) there are very few paths to becoming a citizen without living in the UK for a period of time (at least 5 years continously) and then at that point, you are still not guaranteed the right to stay.
You will need to apply for a student visa and that only permits you to stay for your duration of study and you cannot apply for any benefits while you are here.
You will not be able to apply for a visa or loans until you are are accepted and confirm your place at your chosen university - but it is better to do all of your research in advance of applying.
I commend your desire to study nursing and your interest in the healthcare system in the UK, but make sure you have done full and complete research on the system. The NHS is not flawless at all and has several areas which fall short of the US healthcare system. It's true you have coverage without insurance, but most people supplement their health care with private insurance if they are able due to the lack of resources available throughout the UK. Additionally, healthcare is not equal through the UK, and some regions offer better standards of care than others.
It sounds like you have a lot of passion to study in Glasgow and that is great, but you really need to do alot more research to fully understand everything involved. It's a time consuming and complicated process to move overseas - but it's definitely a worthwhile experience.
I'm definitely not trying to discourage you, but I just want you to be aware of the big picture.
You should definitely use your potential university as a resource. You will not be the first US citizen to study abroad and I am sure they have services in place to assist potential students such as yourself.
Of course your parents don't want you to go, you will be far away and they will miss you. But if you prove to them that you can do all the research and show them how this experience will benefit you, I am sure they will help you in anyway they can.
Feel free to contact me with additional questions.