Unless the U-verse DVR has some kind of data port that's accessible, your best bet is to use something like the Hauppauge HD PVR box to record HD component analog from your DVR's output to the Hauppauge PVR in real time.
The Hauppauge box can record in any of three different flavors of H.264 HD files (M2TS, TS, MP4) in either 720 or 1080, and it ships with a utility to convert the files to other resolutions and formats, including one for the iPhone and other Apple devices. I have the Hauppauge box and it's a very handy little device and it works flawlessly. The connection between the Hauppauge and your computer is USB 2.0. It can also act as an automated DVR for a U-verse or other cable/satellite tuner because it ships with an IR emitter that can set the receive channel on the tuner and do timer-type programmed recordings.
The downside of course is that the recordings would take place in real time, so if you have a big library on your DVR to move, it's going to take some time to transfer. You'd be hard pressed to tell the difference between its analog recordings and those that are purely digital. I play mine back to my HDTV using a WDTV Live media player plugged into my TV's HDMI port and the picture quality is very good.