Having this code in the Bash startup script is not going to be very useful, since you are only going to be able to use it if you are invoking git from within an interactive login shell. What you really want is to overload the git executable that is found in the PATH.

PATH works exactly the same way on Linux as it does on Windows. It contains a list of directories that is searched for executables whenever you enter the name of a program in your terminal. The default path on Linux is something like /usr/local/bin:/usr/bin. What you can see here is that /usr/local/bin is searched before /usr/bin, as such we can place stuff in there to overload other executables.

To know where your current git is installed use command -v git, which should give you /usr/bin/git. Meaning, we can use /usr/local/bin to overload it. So, let us do exactly that.

cat >/tmp/git <<'GIT'
#!/usr/bin/env bash
set -e
if [[ $(pwd -P) =~ ^/mnt/./ ]]
    then exec /mnt/c/Program\ Files/Git/cmd/git.exe "$@"
    else exec /usr/bin/git "$@"
fi
GIT
sudo install /tmp/git /usr/local/bin

That is all, no matter from where you invoke git it will be using your overloaded version going forward. Whenever you really want to us the original just use /usr/bin/git.

You can do exactly the same with many other programs that read things from disk. For instance https://github.com/sharkdp/bat is really awesome, and available cross-platform. Just brew install bat on Linux and choco install bat on Windows and create an overload with exactly the same logic as in the git overload (obviously replacing the paths to the executables), and enjoy a massive speed up.

Just use it as an alias. I use fish, I added:

alias git "git.exe"