You Do Need power plug adapter while traveling from Luxembourg to United Kingdom (UK).

Here is Why You Need An Adapter

The outlet and plug type used in United Kingdom (UK) is G and in Luxembourg are C and F. This is how they look like.

Luxembourg

In Luxembourg, outlets of type C and F are in use.

Type C

Type F

United Kingdom (UK)

In United Kingdom (UK), outlets of type G are in use.

Type G