Re: How long before all the planets re-align to today's alignment?

Basically, never!
... or at least not for many thousands of years.

The question you raised does not have a simple answer.

The orbital periods of the planets all have different values, and so if you wait until some of the planets are close to today's configuration the other ones will not be near their current positions. It would take an extremely long time to return to a position near to today's positions, but that depends on how close "near" represents.

I have been unable to find a calculation on the Web which states how long it takes for the planets return to any particular apparent position.

There is a lot of literature available on the Web about the frequency of planetary alignments. In reading this, you should be careful to note what kinds of alignments are being discussed. During every decade, a few alignments are interesting enough to be reported in the news. Many of these are such that several bright planets (usually Mars, Jupiter, Venus or Saturn) will be seen in the same general direction in the sky. ("General" usually means within 20--30 degrees or so.) Other times, several planets and/or the Sun and Moon will line up so that you can trace a straight line from one to another to another ...

Don Terndrup
Department of Astronomy
Ohio State University