My iPhone 6s has an download and install button for the update, but whenever I click it, it’s just like nothing happened or it won’t update my phone

Assuming your iPhone supports more recent iOS versions than the one currently running on it (a model 6s or 6s+ can run iOS 15.7.9, no further), and assuming you're trying to update your device wirelessly, and bearing in mind that to perform such a procedure you need to connect your device to a reliable and fast Wifi Network with Internet access (no cellular data, unless you use a 5G network, no personal hotspot on your device, no VPN connections are admitted, no Low Power Mode enabled, and I suggest that you keep the device plugged into power during the whole update process), read this support article:--> If your iPhone, iPad or iPod touch won’t update (if you don't have a reliable and fast wifi network, "borrow" one from your relatives/friends/neighbors).

Also, make sure your device is NOT running a Beta operating system version. Should be this case, you need to remove it before installing any official iOS/iPadOS releases, as per these instructions:--> How to uninstall iOS or iPadOS beta software - Apple Support.

DO NOT forget to create a full backup of your device before updating its operating system, and keep in mind that, depending on the available storage on your device and/or on your wifi network speed, it might take 30-60 minutes to complete the entire update process.

If the issue persists or if the iOS version currently running on your device is very old (from your product list, it seems that the device is still running the old iOS 12), try to update it via computer (if you don't have a computer borrow one from your friends/neighbors, or ask for help at an Apple Store or Authorized Service Provider), then learn how to update your device manually if you're using a Mac with macOS Catalina or later, or if you're using a Mac with macOS Mojave or earlier or a Windows PC.

If you still get no help, contact Apple Support directly, using this link:--> Contact Apple for support and service

Regards

Giulio

Do you have an iPhone 6s (as your tag line indicates), or an iPhone 6?

The names differ by only one letter, so it would be easy to mix up the two, but the identification matters when it comes to determining how far you can upgrade iOS.

• An iPhone 6 can upgrade to the latest release of iOS 12 – currently iOS 12.5.7. It cannot run iOS 13 or higher.
• An iPhone 6s can upgrade to the latest release of iOS 15 – currently iOS 15.7.9. It cannot run iOS 16 or higher.