read the sentence like this Thanks very much ---for the invitation--- to ---- cousin Oliver and Becky's wedding. You have been invited to a wedding. the wedding of cousin oliver and becky. So, you have been invited to cousin oliver and beckyś wedding. did that clearify?

Cousin Oliver and Becky, are getting married. Thank you for giving me the invitation letter (thanks to the person who gave you the invitation, it sounds as though you got the invitation from someone other than Oliver or Becky - or even Oliver and Becky as a couple). An example could be, your mother was going to see you for lunch, and she passed the invitation to you, you would say to your mother, "thanks for the invitation to Oliver and Becky's wedding". Does that make sense? And yes, "to" is a preposition before O+B Wedding.

"to cousin" isn't a verb, it's a noun, well here it's more like an adjective. It's describing the relationship between Oliver and Becky, and the speaker. They are cousins. "To" is a preposition and "cousin Oliver and Becky's wedding" is a noun phrase. It's an invitation to go to their wedding.