I learned how to bird's nest my climbing tape (or webbing, such as the
flat 1" tape you'd purchase by the foot at the climbing shop for use
as slings, etc) and forgot how to do it.  I'm hoping someone more
adept with the web than I can find instructions (and, preferably, a
diagram, but either would be brillant) on how to coil flat webbing
into a birds nest.  The end result is a coil of rope that looks as if
it has been woven together, the uniform surface making it easy to
deploy (no snags when you throw it) and store.  Can someone younger
and more computer savy than me take a shot at this?  Thank you.

Do you mean that the tape is wrapped back and forth in a figure-eight
pattern, like you could wind string between thumb and forefinger?
If so, just turn a chair or stool with parallel legs upside down and do it.
The legs must be parallel, so that you can lift your "bird's nest" off.

I hope that is what you want.

Nope; I'm great at flaking, butterflying, and coiling rope.  But this
ends up looking a little like a woven basket.  Thanks for your
suggestions, however.

When I read your question I thought of a knot that I have often used,
but I wouldn't describe it as a "woven basket". This is the knot I
know:

http://www.virtualinks.com/knotlist.htm#CHAIN

It is called a Daisy Chain. Is this your knot.

Hi, Thanks for the suggestion (and the *nice* site on knots), but that
isn't what I'm after.  The finished product is actually compact, very
neat, and actually does look woven.