Ok, I'm trying something. I've mentioned that I'm not reinstalling Twitter on my phone. So I've built a little Heroku App that emails you tweets from people you don't want to miss a tweet from. It runs once a day, and returns just plain text tweets with as little formatting as possible.
It's setup as a Heroku one click deploy, so you should be able to click a button in the readme and deploy your own version if you're interested. Feature suggestions and tweaks welcome.
It's not complicated code, so hit me up with pull requests or issues on Github.