No. You can open the packs you receive as Champions rewards at any time. They don’t expire. You don’t have to claim your rewards before the next round starts. They will be waiting for you as soon as you open UT on your console next time.
If you want, you can save them to open when better items are in packs, as is the case of the Team of the Year. However, remember that Player Picks will always be from the week you have won them.