In answer to another of your questions . .
. . . the fact that you are already coming from a great university that has many of the same resources and courses as HYPSM does not by itself help or hurt your application – it’s a neutral. It’s all about what is on Harvard’s’s transfer applicant web page: https://college.harvard.edu/admissions/application-process/transferring-harvard-college
If you have all three of those elements, and you’ve written that you have . . .
. . . I would think if you have a 3.9+ unweighted college GPA showing that you’ve matured since high school, can identify what academics Harvard would offer you that are not obtainable at The London School Of Economics, and have glowing recommendations from your college professors, then I think you might stand an “average” or slightly “better than average” chance.