You Do Need power plug adapter while traveling from Taiwan 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 Taiwan are A and B. This is how they look like.

Taiwan

In Taiwan, outlets of type A and B are in use.

Type A

Type B

United Kingdom (UK)

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

Type G