All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
As for fasting, whoever wakes up after the time of Fajr has started while he is in the state of Janaabah (sexual defilement), then he fasts and there is no blame on him for that.
The evidence for the above mentioned ruling is the Hadeeth narrated by ‘Aa’ishah and Umm Salamah that “The Prophet would rise in the morning on a fasting day while he is sexually defiled and then he would perform Ghusl (a ritual bath) and fast.” [Al-Bukhari and Muslim]
Imam Muslim added “And he does not make up that day.”
As for the prayer, it is obligatory to perform Ghusl for prayer. If the water is cold, then you should heat it up and perform Ghusl, and it is not permissible for you to opt for Tayammum (dry ablution) instead of Ghusl in this case.
If you do not find anything with which to heat the water and you are afraid that you will get sick if you performed Ghusl with cold water, and you are unable to perform Ghusl with water in a way that does not harm you, then in this case there is no harm on you to perform Tayammum. Therefore, perform Tayammum and pray, and then you perform Ghusl to remove the state of Janaabah after that whenever you are able to perform Ghusl with water for the Hadeeth that reads: “Earth (soil) is a means of ablution for a Muslim, even if he does not find water for ten years; but when he finds water, he should fear Allah and make it touch his skin.” [Al-Baz-Zaar; Al-Albaani: Saheeh]
After performing Ghusl, you are not obliged to repeat the prayer that you had prayed with Tayammum.
Ibn Qudaamah said in Al-Mughni:
“If one fears extreme cold and he can heat the water or use it in a manner that protects him from harm – like washing the body parts (that should be washed in ablution) one after the other and once he washes an organ, he covers it – then he is obliged to do so. If he is unable to do so, then, according to the view of most scholars, he may perform Tayammum and pray.
Moreover, Abu Daawood and Abu Bakr al-Khallaal reported in a chain of narration to ‘Amr ibn al-‘Aas, who said: “I had a wet dream on a cold night in the battle of Dhat as-Salaasil. I was afraid, that if I performed Ghusl I would die. I, therefore, performed Tayammum and led my Companions in the Fajr prayer. They mentioned that to the Prophet who said: “O ‘Amr, you led your Companions in prayer while you were sexually defiled?” I informed him of the cause which impeded me from performing Ghusl. And I said: 'I heard that Allah says: “Do not kill yourselves, verily Allah is Merciful to you”. The Prophet laughed and did not say anything.”
The fact that the Prophet kept silent indicates that it is permissible, because he does not approve an error; and because the person was afraid for himself [his life], so he was permitted to perform Tayammum like a wounded person or a sick one. Now, if one performed Tayammum and prayed, is he obliged to make up his prayer? There are two narrations about this: one narration stipulates that he does not need to repeat the prayer; this is the view of Ath-Thawri, Maalik, Abu Haneefah, and Ibn al-Munthir, for the Hadeeth of ‘Amr as the Prophet did not order him to repeat the prayer. Had it been an obligation to repeat the prayer, the Prophet would have ordered him to repeat it. ‘Amr was afraid for himself, so he was like a sick person, and he did what he was ordered to do [acted according to the verse]; so he was like everyone else who prays with Tayammum.
The second view stipulates that he is obliged to repeat the prayer; this is the view of Abu Yusuf and Muhammad, because it is a rare excuse and not continuous, and does not prevent from performing again the prayer, like forgetting to perform Tahaarah (purity). The first view is the most correct opinion.” [End of quote]
Allah knows best.