What is the adverb in the sentence yesterday Nick visited and old village?

The Words 'Yesterday Nick visited and old village' is NOT a grammatically correct sentence for two reasons. First, the Word 'and' is a connecting Word to make a compound subject, compound verb, or compound sentence. Instead of 'and' in your sentence, you need one of three Words: a, an, or the. Those three Words are called articles of speech and come before nouns. Second, you need a comma after Yesterday. Note: you use 'an' before Words beginning with a vowel.

So the sentence should read:

Yesterday, Nick visited an old village. (or the old village)

Yesterday is an adverb.

Nick is the noun

an old village is the object phrase

an is an article

old is an adjective modifying village

village is the object.

You can also write the sentence: Nick visited an old village yesterday.