Method 1
Go to your UT folder. Program Files (x86)\Steam\steamapps\common\Unreal Tournament)

then go into the System folder there and open up the file UnrealTournament.ini in a text editor.
scroll down to the section called [WinDrv.WindowsClient]
change these 2 values:
FullscreenViewportX=1920
FullscreenViewportY=1080

Save and the game will run perfectly in 1920x1080

If the GUI seems too small for you to read after changing to a higher screen resolution do the following in game:
Preferences > Video > then change Font Size to Double.

This may be simple to most but there was no guide for it so I've added it just incase anyone is stuck.

Method 2
Extra via comments
In game, press the "~" key and type "setres 1920x1080" (without quotations and with whatever resolution you want it at.)

You will still need to go to Preferences > Video > then change Font Size to Double. if you want the GUI to look correct at a higher screen resolution.