When you install updates from the device itself, the download and installation are done in a single operation. It’s a copy of an invisible installer that’s downloaded. In 2017:
How can I see whats downloading on my iPhone?
tap on the Downloads folder if you want to check the files. Where do I get my downloads? The Downloads folder is located in the Files app and can be found at the bottom-right corner.
How do I know if an app is downloading on my iPhone?
In the older software, you can’t see exactly when an app was purchased, but in the latest software, you can go to the App Store and select Update, then click the top right to see if the app was paid or free.