When using WooCommerce or Easy Digital Downloads with Jilt, are you logged in as an administrator for your store? This is the only scenario in which Jilt for WooCommerce/EDD will not restore the cart or automatically log the customer in when clicking a checkout link or button in recovery emails.
Administrators are not automatically logged in with our WooCommerce/EDD plugin for security concerns. Logging a customer in via email is a reasonably safe thing to do (as email is a verified way of getting log in credentials anyway), but logging an administrator in to recover the cart is a riskier behavior since this type of account has access to everything on your site.
We have opted to prevent automatically logging in administrators for this reason, as it's also very unlikely that your customers would have this capability on your site.
How do I test Jilt then?
To test Jilt's connection to your WooCommerce or EDD store, you could create a new user with a Customer role (perhaps using a different email address). You can also test this as a guest user by entering a new email at checkout without an account. Then you could follow the same steps to abandon a cart -- add a product to a clean cart, then wait for the recovery email.
In either case, since these customers are not administrators, the recovery emails will properly restore the cart from its abandoned state, and if you use a registered account, you'll also automatically be logged in by clicking the recovery email link.