Described in the Community Dances Manual, book III as being the English version of "Strip the Willow". In this instance, the "Strip the Willow" is refering to the Scottish dance rather than "Strip the Willow" as a figure.
Here the bold shows the words 'as published', the notes and annotations hopefully make things easier.
The 'next' couple, the one waiting for their go can get going with the single reel when the first couple are out the way. A good guess is when they are half way down the set, you don't want to start too early and risk catching them up (it gets messy). You might also want to wait until the start of the phrase of music.
The strip the willow figure goes back a long way, there's a reasonably clear description in Trenchmore, published in the 2nd Edition of Playford (1652). The third part of which is:
Which is, perhaps, a description of the man stripping he willow down the set followed by the woman stripping the willow up. Trenchmore can be also be traced back quite a bit further