Hi,
The add to cart system will automatically redirect to the checkout if nothing does something after the add to cart.
So, if the cart module updates itself, or the cart notification plugin does something, the add to cart system won't redirect to the checkout.
I suppose you might have the cart module on the page, and that's why there is no redirect.
The solution is actually to activate back the cart notification plugin. It has settings where you can choose to redirect the user to the checkout on success. This will force the redirect after the notification box is displayed by the plugin.