Ubuntu Wifi drivers not working after getting update

I have Ubuntu 16.04 installed on my laptop. It was working fine until one day an update was there. After the update, I am on kernel 4.15.0-13-generic and there it is. My Broadcom Wifi drivers were nowhere to be detected. On Notification panel, there is no option to switch ON/OFF wifi.

result of lspci -nnk | grep 0280 -A3:

02:00.0 Network controller [0280]: Broadcom Limited BCM43142 802.11b/g/n [14e4:4365] (rev 01)
    Subsystem: Foxconn International, Inc. BCM43142 802.11b/g/n [105b:e071]
03:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller [10ec:8168] (rev 0c)
    Subsystem: Sony Corporation RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller [104d:90be]

I tried different kernels on the present in the grub and they are working fine. Currently, I have to switch to kernel 4.4.0-119.143 to use my wifi.

Here is the error output of command sudo apt-get install bcmwl-kernel-source:

Reading package lists... Done
Building dependency tree       
Reading state information... Done
The following NEW packages will be installed:
  bcmwl-kernel-source
0 upgraded, 1 newly installed, 0 to remove and 0 not upgraded.
Need to get 1,544 kB of archives.
After this operation, 8,064 kB of additional disk space will be used.
Get:1 http://in.archive.ubuntu.com/ubuntu xenial-updates/restricted amd64 bcmwl-kernel-source amd64 6.30.223.271+bdcom-0ubuntu1~1.2 [1,544 kB]
Fetched 1,544 kB in 5s (295 kB/s)                
Selecting previously unselected package bcmwl-kernel-source.
(Reading database ... 290899 files and directories currently installed.)
Preparing to unpack .../bcmwl-kernel-source_6.30.223.271+bdcom-0ubuntu1~1.2_amd64.deb ...
Unpacking bcmwl-kernel-source (6.30.223.271+bdcom-0ubuntu1~1.2) ...
Setting up bcmwl-kernel-source (6.30.223.271+bdcom-0ubuntu1~1.2) ...
Loading new bcmwl-6.30.223.271+bdcom DKMS files...
First Installation: checking all kernels...
Building only for 4.15.0-20-generic
Building for architecture x86_64
Building initial module for 4.15.0-20-generic
ERROR: Cannot create report: [Errno 17] File exists: '/var/crash/bcmwl-kernel-source.0.crash'
Error! Bad return status for module build on kernel: 4.15.0-20-generic (x86_64)
Consult /var/lib/dkms/bcmwl/6.30.223.271+bdcom/build/make.log for more information.
modprobe: FATAL: Module wl not found in directory /lib/modules/4.15.0-20-generic
update-initramfs: deferring update (trigger activated)
Processing triggers for initramfs-tools (0.122ubuntu8.11) ...
update-initramfs: Generating /boot/initrd.img-4.15.0-20-generic

Also, tell me if I update my ubuntu 16.04 to ubuntu 18.04 will help to solve this problem.

The version of the Broadcom driver that matches your running kernel 4.15.0-xx is 6.30.223.271+bdcom-0ubuntu4. Because you started with and still have the apt repositories for Ubuntu 16.04, the system tries to install an earlier version; viz:

Unpacking bcmwl-kernel-source (6.30.223.271+bdcom-0ubuntu1~1.2) ...

Please open a terminal and do:

wget http://mirrors.kernel.org/ubuntu/pool/restricted/b/bcmwl/bcmwl-kernel-source_6.30.223.271+bdcom-0ubuntu4_amd64.deb
sudo dpkg -i bcmwl-kernel-source_6.30.223.271+bdcom-0ubuntu4_amd64.deb

Reboot.