MacBook Pro says no internet and won’t update despite Wi-Fi connected

IF you have a different Mac, you can use it to download MacOS install image, then interrupt the process and create a BOOTABLE USB-stick Installer/Utilities stick. BOOTABLE is key, because the way you will install from this USB-Stick is to BOOT the USB-stick, and use its Utilities to ERASE your drive and start the Installer. here is the article on bootable USB-Stick Utilities/Installer:

What you need to create a bootable installer

• A USB flash drive or other secondary volume formatted as GUID partition Map, Mac OS Extended, with at least 14GB of available storage

• A downloaded installer for macOS Big Sur, Catalina, Mojave, High Sierra, or El Capitan.

The Terminal command assumes that Installer in located in the /Applications folder.

from:

How to create a bootable installer for macOS

Create a bootable installer for macOS - Apple Support

NB>> if you name your incoming USB stick exactly MyVolume, you can copy and paste the very long Terminal command from the article directly into the Terminal window, without having to change anything.

10.11 El Capitan is a recommended waypoint, even if you expect to install a later version. 10.11 allows encrypted Internet connections and Mac App Store access. That makes getting later versions much easier.

That Mac and the Mountain Lion OS are both quite obsolete. It can be done, but successfully upgrading it will be a lesson in Mac operations to say the least, even for a veteran Mac user.

What sort of use does your wife hope to get from this computer?

You posted that the mac has been wiped and it is running Mountain Lion. So... does the computer actually boot properly into Mountain Lion? Hopefully so, since that will make upgrading easier. If the computer does not currently boot up properly into Mt. Lion, then things will be more challenging.

If the computer boots to Mt. Lion, then you may be able to download the installer for Mac OS X 10.11 El Capitan and run it. If need be you can use another Mac to download that installer if this MBP won't connect to the internet.

How to download and install macOS - Apple Support

The direct download link for El Capitan is here: El Capitan 10.11

Another optional way to upgrade this computer is to use a newer Mac to download the macOS 10.13 High Sierra installer from the App Store and then process that to create a bootable USB installer. I suggest High Sierra because that is the latest macOS version that the MBP can run.

To do this, you'll be required to use one of these High Sierra compatible Macs:

• MacBook (Late 2009 or newer)
• MacBook Pro (Mid 2010 or newer)
• MacBook Air (Late 2010 or newer)
• Mac mini (Mid 2010 or newer)
• iMac (Late 2009 or newer)
• Mac Pro (Mid 2010 or newer)

(Note - read "or newer" to mean "through late-2015")

Download the macOS 10.13 High Sierra installer from the App Store

Create a bootable installer for macOS - Apple Support

Internet Recovery is NOT available in ROM in most Macs before 2011 models. So you will need to explore this list of other possibilities.

Computers that can be upgraded to use OS X Internet Recovery - Apple Support

When your computer was released, the way you launched the required Utilities (including Disk Utility and Installer) was to use the ones on the Release software DVD. if you have a model-specific version for your model (unlikely) or a Full Retail 10.6 DVD —and a graphic card supported by early 10.6 — you use its Utilities, boot and install that version, then use Software update to get to 10.6.8 with all updates, which is the first version that can reach out to Apple servers and download and install a later version.