I’m having this same problem. I just recently created my first embedded timeline. It shows existing tweets at the time the page is first loaded, but it doesn’t update as new tweets are created. If I refresh the page in the browser, then it updates and shows the more recent tweets. But I need it to do this automatically as refreshing the browser window isn’t an option.
[EDIT]: I refreshed my page (and got the latest tweets into the embedded timeline), then I just made a post of my own on twitter in another tab and the embedded timeline updated. Must be hit-or-miss on whether the embedded timeline tries to update itself.