How can I re-install TikTok on my iPhone

You can't.

TikTok has been banned by the US. government and therefore it will not be available in the US. until the ban has been lifted.

About availability of TikTok and ByteDance Ltd. apps in the United States - Apple Support

Also, read Zachyy's User Tip for more information about this

https://discussions.apple.com/docs/DOC-250009418

Hello~ This is at the top of this page for those that may follow…

💡 Did you know?

TikTok and ByteDance Ltd. apps are no longer available in the United States, and visitors to the United States might have limited access to features. Learn more <<< Click here to view Apple’s current stance.

~Katana-San~

IMPORTANT UPDATE:

The TikTok app has returned to the App Store.

Good luck! 👋🏼😉