New messages will be received on the new iPhone once it has the sim card in.
But existing messages on your old phone will not transfer over just because you insert the sim card in the new iPhone.
If you need to transfer all your existing messages from iPhone to another, you need to backup the old iPhone and restore the backup to the new iPhone.
What type of phone is the new phone you switched to?
If it is not a Apple iPhone, then you would need to de-register your number from iMessage to receive SMS messages on the new phone.
Deregister iMessage on your iPhone or online - Apple Support
The SMS to email is a carrier feature. They would need to contact their carrier and figure out why your messages cannot be delivered to them and are instead sent as email by their carrier.
After you switch the Sim card to a new phone and you don't get the texts from people, and they receive your text as email how do you fix that?
if you take your sim card out put it in a new iphone will your messages be on your new iphone
