ICMP was blocked on AWS. I enabled it.
Changed MTU to 1360 on both server and client.
Also, after some debugging, I got it working somehow (not ideal)
I blocked wireguard from adding any routes by switching off:
Route Allowed IPs (inside wireguard peer settings)
Then ran following:
ip route add 10.1.1.0/24 via 10.1.1.4 dev wg0
Note: 10.1.1.4 is ip address of teltonika router client and 10.1.1.1 is the VPN address of AWS server
Now I can
Though still no sure VPN is using which link (WAN) or (WWAN)
cat /etc/wireguard/wg0.conf file from AWS sever given below