You choose one card name, not one name per opponent.

Alhammarret's second ability happens as Alhammarret enters the battlefield. No one can cast spells or activate abilities between the time a card is named and the time that Alhammarret's last ability starts to work.

Although spells with the chosen name can't be cast, permanent cards with that name can still be put onto the battlefield by a spell or ability.

If your opponents have no nonland cards in their hands, you can't choose a card name. Alhammarret's last ability won't stop any spells from being cast in that case.

Your opponents can still cast a card with the chosen name face down if it has a morph ability. A card with the chosen name can also be manifested.