What ''linking words'' do I use to link these three sentences?

If you mean conjunctions, don't use "and". Use words that show the relationship of the ideas.
Although the official name of the UK is the United Kingdom of Great Britain and Northern Ireland, in actuality, the UK is made up of England, Scotland, Wales and Northern Ireland, while the rest of Ireland is an independent country. 

"is" is the linking verb that is used in the three sentences.