I am currently in my last year of University in the US, studying Political Science.
Is it possible to get accepted to Universities such as UCL, KCL or LSE for a LLM Program, even though I don't have a Bachelor of Law? Is anyone in a similar situation?
Thanks already for any advice.

For the LLM, from the LSE Law Department website LLM FAQ section:

"8. What are the Admission Requirements?

The minimum entry requirement is a 2:1 (or equivalent) in Law, Accounting, Management, Business or Economics.

Your application should include the following;
Two confidential academic references
A transcript of your previous studies
An English language test (TOEFL or IELTS) if applicable
Application fee
Personal statement"

http://www.lse.ac.uk/collections/law/programmes/llm/llm-faqs.htm#7.

As a general proposition, most people on the LLM have a law background. Having said that, LSE, at least, offers an interdisciplinary approach in which, if you are not admitted to the LLM program, there is a Law and Accounting, MSc., if that interests you.

I think a number of institutions accept non-law graduates even without the interdisciplinary approach of the LSE. I think it is a matter of checking websites for the approach of the institution in each case.

Here are some examples: http://www.llm-guide.com/article/213/llm-programs-for-non-law-graduates