I'm ressurecting this old topic since I found out how to do this...
Make sure you are using Aali's OpenGL driver, to use the 'emergency save' feature.
Step 1: Get in the Debug Room.
Step 2: Talk to Yuffie and select 'Global Reset'.
Step 3: Don't press the menu key here or you'll get stuck. Talk to Yuffie again and select 'New Start'
Step 4: The emergency save dialog will pop-up, select yes.
Step 5: Rename crash.ff7 in your save directory to save00-09.ff7
Step 6: Open the save on a save editor (I used Jenova for this) and put yourself on Disk 1. Save the changes.
Step 7: Run FFVII, select the 'Emergency Save', and you'll be at the Debug Room.
Step 8: Talk to Yuffie again and select 'New Start'
Step 9: You should see the beginning FMV. Congrats, you've just started a New Game+!
I don't know what further implications this may have (like for Sephiroth and Young Cloud, since you'll already have at least Cait Sith), but feel free to investigate.