He had a wet dream but he did not do ghusl because it was very cold

It is the mercy of Allah towards His slaves that the obligation to do certain duties no longer applies when one is unable to do them, because Allah says (interpretation of the meaning): 

“Allah burdens not a person beyond his scope” [al-Baqarah 2:286]

This also includes not being able to do ghusl when one is junub (in a state of impurity following sexual activity); whoever is not able to do ghusl because he is sick or it is very cold may do tayammum instead and he does not have to repeat his prayers, because of the hadeeth of ‘Amr ibn al-‘Aas who said: “I had a wet dream on a cold night during the campaign of Dhaat al-Salaasil, and I was scared that if I did ghusl I might die. So I did tayammum and prayed, then in the morning my companions mentioned that to the Prophet (peace and blessings of Allah be upon him). He said, ‘O ‘Amr, you led your companions in prayer when you were junub.’ I told him what had kept me from doing ghusl and I said, ‘I heard that Allah says (interpretation of the meaning): “And do not kill yourselves. Surely, Allah is Most Merciful to you” [al-Nisa’ 4:29].’ The Messenger of Allah (peace and blessings of Allah be upon him) smiled and did not say anything.”

(Narrated by Abu Dawood, 334; classed as saheeh by al-Albaani in Saheeh Abi Dawood, 323; al-Bukhari narrated it as a mu’allaq report). 

If you were in a place where you could not find any means of heating the water and you feared that you might become sick if you did ghusl with cold water, then what you did was correct and you do not have to repeat it.