I am also having the same problem. One one site, I’ve got the CF7 form on a page and it works correctly with AJAX submission and feedback.
On the other site, I’m using the exact same template for the page with CF7, but the form does not even attempt to process with AJAX and goes redirects to the action address of the form, which is “/contact/#wpcf7-f1-p8-o1”.
Any help would be much appreciated.