I'm not too clear on what you are doing here. You transferred your videos to the computer and "it all works flawlessly". What are you using to play the files on this computer?
Transferring them again, to an external drive, and now they don't play at all? Even if you hook the external drive to the original computer and play with whichever media player you used before?
Straight copies to an external drive should not change any of the data.
VLC media player should handle most .mts or .m2ts files with HD video. www.videolan.org If you run into one that VLC cannot play, you could try BSPlayer or Gom Player or Media Player Classic HomeCinema (all free). The commercial program I like best is called Arcsoft TotalMedia Theater 3 Platinum.