Does the new buyer need to worry about the old debtor redeeming the property after he's given it back to the lender via a deed in lieu?

No. By signing a deed in lieu of foreclosure, the seller gave up his right to redeem the property. Furthermore, if the lender had been secured by a deed of trust, the seller (debtor) didn't have an equity of redemption in any event. If the property had been sold under the deed of trust, the sale would have been final.