You Do Need power plug adapter while traveling from Falkland Islands to Tokelau.

Here is Why You Need An Adapter

The outlet and plug type used in Tokelau is I and in Falkland Islands is G. This is how they look like.

Falkland Islands

In Falkland Islands, outlets of type G are in use.

Type G

Tokelau

In Tokelau, outlets of type I are in use.

Type I