I've said before, repeatedly, that you must rename your blog with great care, if it's mature and / or has any reputation. Don't go changing the blog URL without planning the change, if you hope to keep your readers or search engine reputation. That admonition applies, likewise, to a blog with Followers, since people Follow blogs, not people.
If you look at the code behind the "Follow Blog" link in the navbar, you'll see a reference to the "blogID", not the URL. Conversely, when you subscribe to a blog feed, as when you become a Follower of a blog, you subscribe to the URL.
If you look at your Reading List, Google Reader Subscription List, or any other newsfeed client, you'll see blog feeds which are based upon the blog URL. With a new blog name comes a new blog URL, and a new feed URL. Your Followers won't have the updated blog feed in their Reading List, or anywhere else that matters - unless you advise them to change their subscription.
Since the Follower to blog relationship is maintained by blogID, a staged development / test strategy, using a new blog, may not be a good idea. Where a new blog is developed with major changes, your Followers would be left Following a dormant blog.
After you change the blog URL, and the old URL is now free, publish a stub blog to the old URL. Include a notice that the blog has moved, and provide the new URL in the notice, with a clickable link. This will let your Followers and subscribers know to change their subscriptions, and will provide a link for the search engines to follow.
To complement the visual redirect for the blog readers, you can use the Post Feed Redirect URL from the stub, to redirect the blog feed. This will let your Followers and subscribers continue to read the blog feed, using the old feed URL to access the new feed URL.