Sorry for this super bump to the post but, I managed myself to find some of the removed videos.

INFO: Beta Trailer
Broken Link: http://www.gamekyo.com/videofr3657_video-new-super-mario-bros.html
Working Link: http://www.youtube.com/watch?v=WaOJEReEhIo

INFO: Late Beta Video (1:08 Bowser Jr. jumps backwards)
Broken Link: http://www.gamekyo.com/videofr5566_video-new-super-mario-bros.html
Working Link: http://www.youtube.com/watch?v=7BYh8a0qtfo

Other beta videos:

http://www.youtube.com/watch?v=zhocvThdP5g - Very Early Demo
http://www.youtube.com/watch?v=Ie46F2gwBfA - Early Beta
http://www.youtube.com/watch?v=L16AxOA9qgo - Mid-Development Beta
http://www.youtube.com/watch?v=F-AruisRHUY - Pre-Final Beta (Notice the music plays a simple orchestra hit instead of the "Waaa" sound and the "Mario Clear" text at the end of the level is in upper case)