UK universities care mostly about your grades and relevant experience, they tend to gloss over everything else. You'll probably get into LSE, assuming you have the money to pay for tuition and rent. Oxford will be tougher; I'm not sure of your chances because you just about make the cut with an overall GPA of 3.7 but also carry some research experience. It's worth applying, but, keep in mind that Oxford will have a lot of applications from people with 4.0s or the equivalent of 4.0+ and experience, so it will be tough competition.