Changing Emoji Shortcut

This will work everywhere that Emoji & Symbols has a menu item...

• System Prefs > Keyboard > Shortcuts > App Shortcuts
• Click 'All Applications'
  _{(hidden in the picture)}
• Click +
• Type Emoji & Symbols in the first box
• Type your replacement trigger in the second
• Click Add

^{As noted in comments, Cmd/Space is usually used for Spotlight. I chose this almost at random & discarded after the picture was taken.}

As a quick add-on to Tetsujin’s great answer, you could use Karabiner Elements to remap it to a single key if you wished.

Currently I’ve reassigned Caps Lock to showing the emoji picker.