Well, actually, it's not quite so cut and dried. The Feeds API is not deprecated, but there is a notice on the TOS indicating that the deprecation policy will not apply after April 2015. The deprecation policy stated that Google would do its best to maintain the service for three years after making the announcement. The voiding of this portion of the policy means that they may - or may not - shutter the service, with or without warning, at any time after that time.
So I don't think I would spend too much time or energy developing a new application on the back of the Feeds API.
There has been no announcement (that I've seen) of a successor from Google, but there are a couple of viable options. The first option would be Yahoo Pipes. Pipes allows you to select one or more feeds, aggregate them, and output them in a variety of formats, including JSON (which is what you would want to use with JavaScript). And the second would be to build your own feed fetcher/parser/normalizer/cache system and then a REST-based service to serve the content up to your web app.
jg