Learn French: Le délégué parle en tant qu'élu. - The delegate speaks as an elected official.

Just like when stating a profession with the verb être (e.g., Il est médecin), you drop the indefinite article (un or une) after en tant que when describing someone's role, status, or profession.

Is élu a noun or a verb here? It looks like the past participle of élire.

It is the past participle of élire (to elect), but here it functions entirely as a noun. French frequently turns past participles into nouns to describe a person who has undergone an action, so un élu simply means "an elected official."

Could I use comme instead of en tant que in this sentence?

You could use comme élu, but en tant que is better for emphasizing someone's official capacity. Be careful: if you add the article and say comme un élu, it changes the meaning to a comparison ("he speaks like an elected official"), which implies he might not actually be one!

What is the nuance in meaning between a délégué and an élu here?

A délégué is anyone designated to represent a group (like a union representative or a conference delegate), whereas an élu is specifically someone chosen by voters in an election. This sentence sets up a situation where the speaker holds multiple roles but is strictly using their authority as a voter-elected official to make their current point.