Well, you will not be able to have a WILD when you go to sleep in the evening, because it takes around 90 minutes to reach the first REM cycle. But, when you do a WILD, it won't guarantee that you will be lucid in the next REM cycle, too, assuming that you don't do a WILD then.
You don't have to be in REM sleep to WILD.
And in answer to the original questions I doubt your lucid dream would last long enough to last through the transitional stages of sleep. You go from REM to non-REM anyway, as I understand it you are saying from REM to REM - which doesn't happen.
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