I won an auction — how do I get the NFT?

Claim it. Go to Account → Won auctions, open the item and confirm the claim in your wallet. That moves the NFT out of the auction contract and into your wallet, and you pay the network fee for it.

Until you claim, the item is yours but not in your wallet — which is why it does not appear under My NFTs yet. Nothing expires; the claim is waiting whenever you get to it.