1) Avoid jQuery Mobile. It will cause more trouble than its worth. Look at Ionic, Framework7, and similar frameworks instead. Given your mention of jQuery Mobile, Framework7 may prove to be the most comfortable for you.
2) What CMS were you thinking of using? Wordpress has an API you can interface with. Your app would consist of local code that produces the UI, and then uses XMLHttpRequest to connect to your CMS and then pulls down the data. Be sure to introduce some sort of caching mechanism so your app can function offline as well.
Have you considered Ionic framework? We have used both JQuery Mobile, as well as Ionic and happy with Ionic. Ionic is on Angular, so if you know Angular JS, it would be helpful.
If you really need to dig deep, then you would need to go native on iOS and Android for anything that is not available out of the box. We built plugins in the past, and got everything work that is possible in native.
Thank You for your answer, I am really interested in the Wordpress API, I was trying to go through the documentation, but it was confusing. Is there a link that you could recommend in order to get started.