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The OK Computer tour documentary Meeting People Is Easy features an instrumental track that sounds like it could well be an early musical sketch for what would become 'A Rat's Nest', but this connection is purely speculative:
The line "Clicks on the phone" can be seen in this text, that appeared in the artwork of the Special Edition of Hail To The Thief:

Three items from the scrapbook section of are related to this song:

The song is built around a backwards piano track. What sounds like "hello" at the beginning is actually a backwards comment ("voila!") from the end of the original piano track. You can listen to the first 40 or so seconds of the song reversed here:
'A Rat's Nest' was released as a b-side on the Analyse single on november 6th 2006. It was also included on the Spitting Feathers EP, released on november 22nd 2006 in Japan.
On August 5th 2008, Jonny posted a reworking of 'A Rat's Nest' on his page at w.a.s.t.e. central. It uses Thom's original vocal and harmonizes it with new piano chords. He also added sound effects and recordings made from an answerphone: