A diplomatic passport does not automatically entitle the holder to study in the UK without immigration permission. A diplomatic passport only entitles the holder to enter the UK on the same basis as Standard Visitor unless they are posted to the UK for diplomatic purposes.
If you are studying a full time degree at LSE and are not posted to the UK, you will require immigration permission to study in the UK.
By enrolling on a diplomatic passport for a full degree programme you are confirming that: