I don't understand, why you don't want to set the $HOME environment variable since that solves exactly what you're asking for.
cd ~ doesn't mean change to the root directory, but change to the user's home directory, which is set by the $HOME environment variable.
Edit C:\Program Files (x86)\Git\etc\profile and set $HOME variable to whatever you want (add it if it's not there). A good place could be for example right after a condition commented by # Set up USER's home directory. It must be in the MinGW format, for example:
HOME=/c/my/custom/home
Save it, open Git Bash and execute cd ~. You should be in a directory /c/my/custom/home now.
Everything that accesses the user's profile should go into this directory instead of your Windows' profile on a network drive.
Note: C:\Program Files (x86)\Git\etc\profile is shared by all users, so if the machine is used by multiple users, it's a good idea to set the $HOME dynamically:
HOME=/c/Users/$USERNAME
Set the environment variable HOME in Windows to whatever directory you want. In this case, you have to set it in Windows path format (with backslashes, e.g. c:\my\custom\home), Git Bash will load it and convert it to its format.
If you want to change the home directory for all users on your machine, set it as a system environment variable, where you can use for example %USERNAME% variable so every user will have his own home directory, for example:
HOME=c:\custom\home\%USERNAME%
If you want to change the home directory just for yourself, set it as a user environment variable, so other users won't be affected. In this case, you can simply hard-code the whole path:
HOME=c:\my\custom\home