(Not a very practical answer, but maybe can get the ball rolling...)
If you can create a list Youtube links to the songs or albums you want, you can download the captions (subtitles). These have a timestamp along with the lyrics.
With YouTube you can search for videos with captions. Unfortunately, I found that many are listed with captions but actually don't have any meaningful text.

Here is an example of one video with good captions (link).

You can see the transcript by going to "More" and then "Transcript".

One tool to do download captions is called Google2SRT.
What is Google2SRT? Google2SRT is a tool that can download "not embedded" subtitles (Closed Captions - CC) from YouTube and former Google Video and convert them to a standard format (SubRip - SRT) supported by most video players.
There are some online tools, too, such as keepsubs.com and yousub.net (although questionable how reliable).
Also, a Firefox add-on.
You could also use OpenSubtitles.org to download subtitles soundtracks or concerts.