Conceptually, I really like JWT as it is in line with the statelessness of REST etc (no state saved server-side, all relevant data is contained in the token).
What I am
You need to persist the JWT on the client so that it’s available across page loads, the most secure strategy is an HTTPS-only cookie. This will send the JWT to your server on every request and the server can check the validity of the token and reject it if it's expired. How you handle the expiration is dependent on the type of web app you have.
For a single-page application (e.g. Angular.js apps) you would want to structure the application so that it makes an initial request of the server before it bootstraps the rest of the application. If the server sees that the JWT in this request is expired it would issue a 401 response. You application would respond to this response by rendering a login form. Otherwise it would continue with the assumption that the JWT is valid and can be used to access the required resources. If, at any time, the app sees a 401 it should bring the user back to the login form.
For traditional web apps that render their pages on the server: for any request that has an expired JWT (as read from the cookie) the server should issue a 302 redirect to a login form.