Witr can be prayed after 'ishaa up until dawn. So you may pray it after you pray 'ishaa or anytime during the night as long as it is before dawn. So if you have not prayed it at all during the night yet, you may do so after tahajjud, but it is not specifically prayed after tahajjud. And if your wudhu (ablution) is still valid then you do not have to do wudhu again. It is only if you break your wudhu that you will need to redo it for any prayers.
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