Please correct my statements.

The first is right. The second is not. By the way, it is "London, in the UK".

If I am not mistaken, If there are two place in a sentence, That time we use "in" before big place & "at" before the small place.
e.g I live at Paris in France.
When should we use "at" And "in" before place? Thanks

I think you should use "in" in both of your example. I am in London in the UK. (you don't need the dots after U and K)

You can also say "I am in the UK, in London." Stating the country first and then the city sounds better in my opinion.

"I live at Paris in France" is wrong because "in" is the preposition used for both countries and cities.

"

You live, or are, in a town and in a country.
You live, or are, at an address or location.
I live at 123 Some Street, Anytown.
I am at the grocery store in Anytown.