I had a year free of Apple TV that was extended to 2/1/2021. I am in my sister's Apple Music Family account. She just switched to Apple One; how do I NOT get billed on 2/1/2021 without screwing that up?

If you had access to Apple TV+ due to a subscription of one of your Family Sharing group members, then that will expire normally. The person that originally had the subscription will be the one who decides to continue or not.

Apple Music Family is unrelated to Apple TV+.

If your sister has Apple One Individual, then none of the components will be shareable.

If your sister has Apple One Family, then Apple TV+ will be accessible to all members of her Family Sharing group.

If you had subscribed to Apple TV+ for yourself, and want to continue on your sister’s subscription, then you can cancel that personal subscription during the overlapping period. See or cancel your subscriptions - Apple Support

Note that some Apple TV+ subscriptions have been extended once more. So check the new expire date for the trial period in your subscriptions overview.