Why won't my 2015 MacBook Air power up even after charging?

Good job, glad you got it powered on.

As for the OS install, it is hard to say. I've had some installs take a long time and once in a while one will get stuck. A lot of it depends on the quality & speed of your network & local Internet, plus sometimes Apple's servers can be slow if you are still on the downloading phase of the install. Unfortunately there is no easy way to tell if it is still slowly moving or whether it has completely stalled out. If you wait 2 - 3 hours, then I would force a restart and see if it will boot to the internal SSD. If the installer got stuck at the reboot phase, then this may allow it to finish. If the installer already rebooted and got stuck during phase 2, then maybe it will finish, or it may indicate a hardware issue or possibly a file system issue depending whether you erased the drive or reinstalled over top of itself.

Normally I would recommend first creating a bootable macOS USB installer to have on hand just so you have a second option, but if you did not create one previously, then you will need access to another Mac which is compatible with an OS this 2015 laptop supports (10.11 to 12.x) although the other Mac can currently be running any version of macOS.

• For macOS 12.x you generally need a Mac from 2015 to 2021
• For macOS 10.15 you generally need a Mac from 2012 to 2019
• For macOS 10.13 you generally need a Mac from Late-2009 to mid-2018
• For macOS 10.11 you generally need a Mac from 2007 to 2015

You can use the information in the following article to confirm OS & hardware compatibility:

https://eshop.macsales.com/guides/Mac_OS_X_Compatibility

Try an SMC Reset to see if it will allow the battery to charge & the laptop to power on.

Otherwise you will need to disconnect the battery from the Logic Board. Then connect the power adapter & power on the laptop with the battery disconnected. If this works, then reconnect the battery while the laptop is still powered on & let the battery charge to at least 20%, otherwise you may damage the battery if the charger is disconnected. You must be very careful not to allow any of the internal components from making contact with any metallic or conductive objects or surfaces while the Bottom Case is removed or you will damage the laptop. I personally find setting the laptop on its right side with the display open is easiest & safest way to do this. Sometimes I will place the laptop on the back of its display with the keyboard/palmrest open in a vertical position.

Even if this works and allows the laptop to power on & charge the battery, the battery may already have suffered permanent damage for being completely discharged/drained for such a long time. There is risk of the battery swelling.