Here are instructions for HP Stream using live disc and bcm43142 that solved this issue -
https://ubuntuforums.org/showthread.php?t=2319377
Edit -
I understand now, you have tried to download and install the driver using "apt-get" , but you have no ethernet internet connection and so it was not installed.
Also it may not remain after rebooting on a live usb, unless you made a "USB stick with persistence" etc.
But you can make another usb backup iso to usb once done and then it should be saved, your settings.
You will need to gain temporary access to internet, can you get a usb wifi dongle from a friend, neighbor, colleague, to use to download the driver using the same command you used initially ?
https://ubuntuforums.org/showthread.php?t=2319377
Edit -
I understand now, you have tried to download and install the driver using "apt-get" , but you have no ethernet internet connection and so it was not installed.
Also it may not remain after rebooting on a live usb, unless you made a "USB stick with persistence" etc.
But you can make another usb backup iso to usb once done and then it should be saved, your settings.
You will need to gain temporary access to internet, can you get a usb wifi dongle from a friend, neighbor, colleague, to use to download the driver using the same command you used initially ?