Linux Mint

• ping or mtr your router and verify there are no dropped packets (ex: ping 192.168.0.1 or mtr 192.168.0.1 where 192.168.0.1 is your router's IP)
• ping or mtr a public IP (ex: Cloudflare's DNS server: ping 1.1.1.1)
• ping or mtr a fqdn (ping www.google.com)

If you can ping your local router with no dropped packets, your local network is probably fine

If you can't ping 1.1.1.1, then there is an issue between your router and the Internet. Look at the output of mtr to see if/where there is packet loss

If you can ping 1.1.1.1, but not www.google.com then your issue may be DNS. Verify your DNS server IPs

*If you don't know your router's IP, try 'route -n' and the default gateway should be the non 0.0.0.0 entry under the Gateway column, usually it's in the first line

See the network icon on the right-hand side of the bottom panel? What does it show when you click it?

• Is there a "Wireless" section?
• Does it have an activated checkbox next to it?
• Is there a little circle to the left of your wireless network name?
• And what percentage is showing for the link strength?

If the "Wireless" section is missing, go to the main Mint menu, run "Driver Manager" (under "Administration"), and see if it finds a WLAN driver for you. A working connection in the live environment isn't necessarily transferred into the installation.

this sounds more like network problem than a linux one; but since you're using linux w/o an internet connection, you'll have to use the tools that usually come with Linux:

get a sense of your computer's situation some of these commands will give you the same information and i write them here because i don't know what tools mint provides with its image:

start with basic information. see if your computer has an ip address and who its primary gateway(s) is/are:

ip a
ifconfig -a
netstat -rn
route -n

then use that information to start troubleshooting; these commands are a shortcut; duplicate information; and should be run in order:

for i in $(netstat -rn | grep ^0.0.0.0 | awk '{print $2}') ; do ping -c 3 -W 1 $i ; done
if [ $? -eq 0 ] ; then cat /etc/resolv.conf ; traceroute google.com ; fi

this command provides duplicate information but provided here in case mint has really weird packages on it

for i in $(route -n | grep ^0.0.0.0 | awk '{print $2}') ; do ping -c 3 -W 1 $i ; done
if [ $? -eq 0 ] ; then cat /etc/resolv.conf ; traceroute google.com ; fi

if neither of the last 2 stanzas worked, then your computer doesn't think it's connected to a network and the source of your problem is somewhere with either the hardware, driver or mint.

Can you ping IPs (like your router or 1.1.1.1)? Do you have the same behavior over ethernet?

Also, might be silly bit I’ve been fooled by this more once, have you tested other devices have access to the internet and the specific sites you’re testing with aren’t down?