If one has been praying Salatul Witr as wajib due to having taken an oath and comes in to a situation when they wake up that there is only a minute remaining. How will the person make sure he completes his prayer within the time. E.g in a normal wajib prayer due to lack of time he would not pray the second surah , leave the qunut etc. How will he do this for his Salatul witr?

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You can do Fajr then pray witr as qadha. Salaatul Witr is mustahab and all the parts of it are mustahab.

So you have two options either you can do Salaatul Witr with just Fatiha and Ikhlas, Ruku and sujood if there’s time for just that or you can do Fajr and then pray Witr as qadha