how do I update WhatsApp on my Mac and my iPhone if I deleted the old Apple ID

Yes, but unfortunately in deleting an app you lose the data associated with the app.

It may be possible to do a workaround by offloading the app instead. This removes the app from your device but keeps the data. You could then download the app again and maybe it will do this from the account to which you are signed in.

I have never done this so I really don't know if that workaround will actually work or if there may be other issues related to the data perhaps being linked to an Apple ID and cannot be re-used with a different one. Perhaps somebody who has encountered this issue may know.

Offload app

https://support.apple.com/HT201656

> Use your device to check its storage

Offload the app, which frees up storage used by the app, but keeps its documents and data.

Delete the app, which removes the app and its related data.

Redownload the app with your new Apple account. You cannot update an app except with the account that was used to download it.