The Ministry only takes wands from the criminally charged, not students who leave Hogwarts.
It seems likely that the reason Hagrid had his wand snapped was because of his suspected involvement in the death of a student because he kept an Acromantula at Hogwarts, not because he was expelled from Hogwarts. His expulsion and the confiscation of his wand were both caused by that incident. The Ministry cannot confiscate wands until charges are successfully proven, and while Hagrid’s charges were not proven, he was suspected of being involved in a crime by negligence.
‘The Ministry does not have the power to expel Hogwarts students, Cornelius, as I reminded you on the night of the second of August,’ said Dumbledore. ‘Nor does it have the right to confiscate wands until charges have been successfully proven; again, as I reminded you on the night of the second of August. In your admirable haste to ensure that the law is upheld, you appear, inadvertently I am sure, to have overlooked a few laws yourself.”
- Harry Potter and the Order of the Phoenix, Chapter 8 (The Hearing)
Fred and George were never charged with anything - they just chose to leave Hogwarts in their sixth year. Attending Hogwarts was never compulsory, except when Voldemort took control of it, so the Ministry could not take wands from anyone not attending or not finishing Hogwarts.
“What’s Voldemort planning for Hogwarts?’ she asked Lupin.
‘Attendance is now compulsory for every young witch and wizard,’ he replied. ‘That was announced yesterday. It’s a change, because it was never obligatory before. Of course, nearly every witch and wizard in Britain has been educated at Hogwarts, but their parents had the right to teach them at home or send them abroad if they preferred. This way, Voldemort will have the whole wizarding population under his eye from a young age.”
- Harry Potter and the Deathly Hallows, Chapter 11 (The Bribe)
Therefore, students who leave Hogwarts without having any criminal charges would not have their wands taken.
Newt was never actually expelled, he was cleared and never charged.
A MACUSA document in an official companion book shows that Newt was never actually expelled from Hogwarts. Dumbledore defended him and his name was cleared before he was expelled, so he would have been able to continue attending Hogwarts and his wand would not have been broken.
NOTES: Proposed expulsion from Hogwarts School of Witchcraft and Wizardry due to the illegal possession of a magical beast causing endangerment of human life and violation of animal welfare laws 101/304.
Magical Beast: Jarvey, Level 3
Punishment: Expulsion from Hogwarts School of Witchcraft and Wizardry.
– The expulsion was never enforced - Hogwarts Professor Albus Dumbledore defended Newton resulting in his name being cleared.
– The Ministry of Magic Animal Welfare Department were also informed of the violation and required to rehouse the magical creature.
-The Case of Beasts: Explore the Film Wizardry of Fantastic Beasts and Where to Find Them
This is similar to when Harry was almost expelled, but was never actually expelled due to Dumbledore’s defense of him. Harry had gotten very close to being expelled. He received a letter from the Ministry claiming he was expelled, which also mentioned destruction of his wand.
“We have received intelligence that you performed the Patronus Charm at twenty-three minutes past nine this evening in a Muggle inhabited area and in the presence of a Muggle.
The severity of this breach of the Decree for the Reasonable Restriction of Underage Sorcery has resulted in your expulsion from Hogwarts School of Witchcraft and Wizardry. Ministry representatives will be calling at your place of residence shortly to destroy your wand.”
- Harry Potter and the Order of the Phoenix, Chapter 2 (A Peck of Owls)
But, since Dumbledore’s defense of him, Harry had been proven as innocent. Because of this, he was never expelled, so his attendance at Hogwarts continued as normal. Dumbledore also reminded the Minister of Magic that the Ministry cannot confiscate wands until charges were successfully proven. The charges against Newt also had never been proven, so the Ministry could not destroy his wand.