Hello -Captain-,
We understand that your sister has an iPhone that is receiving text messages later than anticipated. If you have her iPhone handy, let's work to isolate the situation.
Does this happen with SMS messages, iMessages or both? How about Wi-Fi vs cellular data connectivity? Is she having any issues with sending messages? Is your sister's iPhone up to date running iOS 14.4.2? If not, please be sure she backs up her data first, then install the update.
How to back up your iPhone, iPad, and iPod touch
Update your iPhone, iPad, or iPod touch
If you are going to be taking over her iPhone, it will reason that you'll want to remove all her data from the device and at that time you can restore the software to be sure there isn't an issue there, since it won't contain any of your data yet. Then restore your backup or migrate your data to the device. Restore your iPhone, iPad, or iPod to factory settings
Let us know the above answers or if you have any questions, we'll be glad to keep working on this with you and your sister.