I have the exact configuration - 2010 MBP with Nvidia Gforce 320M but DE - Gnome
I removed the Drivers - broadcom-wl instead i have installed broadcom-wl-dkms.
$ sudo pacman -Sy broadcom-wl-dkms
$ sudo modprobe wl
Usually when you use the above commands - it should work, but for me I got an error -
" modprobe: FATAL: Module wl not found in directory /lib/modules/5.4.89-1-MANJARO"
you would have to install linux headers, I have 5.4 and 5.10 kernels, so I have installed corresponding linux headers again.
$ sudo pacman -Sy linux510-headers
$ sudo pacman -Sy linux54-headers
Now, I have wifi working for the Kernel v5.4 only
But for v5.10, I could see the menu “wi-fi not connected” in the top right corner and it does not list any wifi connections.
I will try to create a bug report for - “v5.10 broadcom-wl-dkms not working”.
Hopefully the above steps should work for you, I would recommend to use kernel v5.4 only. I have had issues with v5.8, v5.9 as well in the past for this Hardware config.
Hopefully it works for you as well.