Upvote:1
While there are circumstances under which you might apply to extend a Tier 5 visa, it may not be the most effective way to go about remaining in the UK.
It is extremely ill-advised to stay past the end date; it would constitute an overstay in your immigration record and, in future, you would need to apply for a UK visit visa, rather than allowed visa-free entry as a US citizen.
You should leave the UK by the end of your visa validity, as you already plan to do to attend the wedding in Luxembourg. On your return, you present at the border as a US citizen seeking visa-free entry. You may not be asked, but be prepared to explain your plans, to retrieve your belongings and take your flight home.