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I prayed qiyam in the mosque and afterwards, the imam started to pray Witr with three continuous rak’ahs, but I had come late and had missed ‘Isha.
Praise be to Allah.
Qiyam and Witr can only be prayed after praying ‘Isha. So you should have prayed ‘Isha first and then prayed qiyam and Witr after that.
Al-Nawawi said in al-Majmu’ (3/526):
“The time for Tarawih begins when ‘Isha prayer is over, as was stated by al-Baghawi and others, and lasts until dawn comes.”
It says in al-Insaf (4/166):
“The time for it (i.e., Tarawih) begins after ‘Isha prayer and its Sunnahs, according to the correct view. This is the view of the majority and is the practice of the Muslims.”
Shaykh Ibn ‘Uthaymin said:
“The time for Tarawih is from after ‘Isha prayer until dawn comes.” Majmu’ Fatawa Ibn ‘Uthaymeen, 14/210
Based on this, what should a person do if he comes to the mosque late, when the imam has already finished ‘Isha and has started Tarawih?
The correct view is that he should join the imam, with the intention of praying ‘Isha, and he should stand to complete his prayer (i.e., four rak’ahs) after the imam says the salam. Then he should join him in the remainder of the Tarawih prayer.”
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) said:
There is nothing wrong with praying ‘Isha behind one who is praying Tarawih . Imam Ahmad (may Allah have mercy on him) stated that if a man enters the mosque in Ramadan when they are praying Tarawih, he should pray behind the imam with the intention of praying ‘Isha, then when the imam says the salam at the end of the prayer, he should complete whatever remains of ‘Isha prayer.” (Majmu’ Fatawa Ibn ‘Uthaymin, 12/443-445)
For more about praying Tarawih before ‘Isha, please see this answer: 37829
And Allah knows best.