You don't need to remove Nautilus - just set Nemo as default - see here.
To install Nemo, open a terminal.
sudo add-apt-repository ppa:webupd8team/nemo
sudo apt-get update
sudo apt-get install nemo nemo-filerolle
If you need to install extensions for Nemo (optional)
sudo apt-get install nemo-compare nemo-dropbox nemo-media-columns nemo-pastebin nemo-seahorse nemo-share
To use Nemo as default File Manager, you will need dconf-tools
sudo apt-get install dconf-tools
To disable Nautilus from drawing the desktop item,
gsettings set org.gnome.desktop.background show-desktop-icons false
To Set Nemo as the default file manager (replacing Nautilus),
xdg-mime default nemo.desktop inode/directory application/x-gnome-saved-search
If you have removed unity, you should be able to reinstall it.
Start your computer and login & open a terminal.
If you can't login, enter the tty with Ctrl+Alt+F2.
Then run:
sudo apt-get install unity-2d
sudo apt-get install ubuntu-desktop
sudo apt-get install ubuntu-desktop-2d
sudo apt-get install compizconfig-settings-manager
sudo apt-get install xserver-xgl
sudo apt-get install emerald
sudo apt-get install compiz-fusion-plugins-extra
sudo apt-get install git compiz-plugins-extra
sudo apt-get install compiz-plugins-extra
sudo apt-get install unity
Then reboot - if in the tty just run:
sudo shutdown -r now