I would try duplicating this site to a staging area.
Then go through and deactivate all of your plugins except for Woo Commerce. With everything disabled except Woo Commerce and your theme, try and add items to your cart. See if there is a delay for things when you add them to the cart.
If there is no delay, slowly activate your plugins one at a time until your find the plugin that is causing the issues. If there is a delay with only your theme and Woo Commerce, then try switching to a different theme.
This way you can slowly narrow things down to determine what is at fault.