The government said it expected schools to be "phone-free by default" as a result of the announcement.
The government says the industry will have nowhere to hide, but campaigners say the reforms do not go far enough.
Sir Keir Starmer said the situation was "very serious" and that right approach was through "calm discussion".
The prime minister is seeking to downplay how he intends to react to Trump's threat of tariffs.
The prince is one of seven high-profile claimants that allege the publisher of the Daily Mail and Mail on Sunday committed "grave breaches of privacy".