I have US and UK apple accounts. My ipad is currently setup for the UK with my UK account.
If I wipe the ipad, switch the region to the US and login with my US account.
Will I also be able to login to the Prime video app with my UK Amazon account?
You don’t need to wipe the iPad if you just want to switch to your US Apple account. You can do that as many times as you like and it won’t check to make sure you’re in the country. However, if you authorise it within iTunes to play purchased video, I think you can only do that once ever 90 days (or something restrictive like that).
I don’t know if Prime checks your IP address, it might not even bother. If it does, just download a VPN first and switch country.
Logging in and out of 3rd party apps in different ”countries” with different accounts won’t do any harm - they’ll either work or they won’t. They also don’t care what “format” your Apple account is in.