To fix this problem you need to do is go in to the registry for the bottle you've installed your game, and add the registry key
HKEY_CURRENT_USER -> Software -> Wine ->DirectInput ->MouseWarpOverride and give it the value 'force'.
To do that, you go to the CXG Config screen, Manage Bottles, click the bottle with the game in, and click Configure. Go the Control Panel tab, click 'Run...' and then type 'regedit'.
Navigate to HKey_Current_User -> Software -> Wine, then rightclick Wine and create a New Key. Call the key DirectInput. Left click the DirectInput you just made so it's highlighted, then right click in the right hand side pane. Create a new String value called MouseWarpOverride. Double click the MouseWarpOverride you just made, and in the 'value data' box type in 'force' without the quotation marks. Close reedit. Make sure all the bottles are stopped, and then try your game again - with any luck you'll have a fully working UTIII.