Do I need special visa from the UK embassy?
No, you need an ordinary Standard Visitor visa, which you must apply for in advance at a visa application centre that covers the country you live in. Visas cannot be gotten at the border.
Since there is only two weeks until you leave, you would probably need to pay extra for premium service (both to get an appointment at the visa centre quickly enough, and for priority processing of the application afterwards), if you want any reasonable chance of getting your passport back before you need it.
As for taking the train to Edinburgh, be aware that Paris to Edinburgh by train will take you close to 8 hours and it is quite likely that flying will be cheaper even for direct flights (about 1h45 plus airport overhead at each end).