There are apps availanle for each phone oS, that you can try, but none is working as good and as smooth as web version of Redmine. RedmineUP Cloud comes with a fully responsive design, which allows you to use all the features on smartphone and tablet.  

You can test the mobile apps to check status of tasks or report time on the go, but for a compatibility details you have to contact an app developers. 

For iOS / iPhone

• Kever simple app with custom dashboard and reporting
• RedminePM for iPhone/iPad is an iPhone/iPad client for Redmine. Multilingual - English, Japanese, German, Russian, Spanish, Portuguese, Chinese, French, Italian, Turkish and Korean languages support.
• BugBranch iPhone and iPad client for Redmine, Jira, Trac, Subversion, Mercurial, Git, and more.
• Redmine Mobile - you can view, create and edit issues and projects, keep track of activities using Redmine Mobile. 
• HiveMine A clean and simple Redmine client

For Android

• Redminer - cool for controling the work progress and reporting time when away from the computer,
• MyMine: Redmine for Android is a Holo designed Android client supporting multi servers, multi projects, issue editing, wiki and much more
• OpenRedmine open-sourced app under GPLv2+ on Github
• RedminePM for Android multilingual client for Redmine
• RedMiner Droid another app

For Windows Phone

• Redmine Client 
• Redmine To Go - offers offline editing and wiki browser