Praise be to Allah.

Firstly: 

Staying overnight in Muzdalifah is obligatory according to the majority of scholars; some of them are of the view that it is a pillar or essential part of Hajj. 

There was a difference of opinion as to how long one must stay there. The view of the Shaafa‘is and Hanbalis is that staying in Muzdalifah is obligatory, even if it is only for a moment, so long as that occurs in the second half of the night, after standing in ‘Arafah, and it is not essential to stay there; rather it is sufficient to simply pass through it. 

So if you had entered within the boundaries of Muzdalifah, even for a moment, between halfway through the night and dawn, then you have done what is required. 

But if you were not able to enter Muzdalifah because the roads were so crowded, then you do not have to do anything (to compensate for that), because obligatory duties are waived when one is unable to do them. Allah, may He be exalted, says (interpretation of the meaning): “On no soul does Allah place a burden greater than it can bear” [al-Baqarah 2:286]. 

Shaykh Ibn Baaz (may Allah have mercy on him) was asked: We were not able to stay overnight in Muzdalifah because we could not find any space except on the road, and they do not allow anyone to stop in the road, so we left and went to Mina. Do we have to do anything (to compensate for that)? 

He replied:

If there was no space in Muzdalifah or the troops did not let him stop, then he does not have to do anything (to compensate for that), because Allah, may He be glorified, says (interpretation of the meaning): “So keep your duty to Allah and fear Him as much as you can” [at-Taghaabun 64:16]. But if that was due to carelessness on his part, then he has to offer a sacrifice and also repent.

End quote from Fataawa Ibn Baaz (17/287) 

See also the answer to question no. 14632 

Secondly: 

You made a serious mistake by delaying the prayer until the time for it was over, because that is haraam and is in fact a major sin. Allah, may He be exalted, says (interpretation of the meaning):

“Verily, the prayer is enjoined on the believers at fixed hours”

[an-Nisa’ 4:103]

“Then, there has succeeded them a posterity who have given up As-Salat (the prayers) (i.e. made their Salat (prayers) to be lost, either by not offering them or by not offering them perfectly or by not offering them in their proper fixed times, etc.) and have followed lusts. So they will be thrown in Hell”

[Maryam 19:59]. 

The Sunnah is to delay Maghrib and put it together with ‘Isha’ in Muzdalifah, but if a person is afraid that the time for ‘Isha’ will end before he reaches Muzdalifah, then he has to pray on the road, according to his situation. If he is walking, he should stop and pray, standing, bowing and prostrating; if he is riding (in a vehicle) and cannot get off, he should pray on board that vehicle, because Allah, may He be exalted, says (interpretation of the meaning): “So keep your duty to Allah and fear Him as much as you can” [at-Taghaabun 64:16]. 

If a person remains in ‘Arafah until approximately 10 PM, and is aware that the roads are crowded, it is better for him for him to pray Maghrib and ‘Isha’ in ‘Arafah, especially if he has women with him, for whom it is difficult to get off and pray in the road. 

The point is that one should not neglect what is obligatory for the sake of something that is Sunnah. 

What you have to do is repent to Allah, may He be exalted, for having delayed the prayer, but you do not have to do anything else. 

And Allah knows best.