What is the ruling on the fast of one who intended to fast, then ate the pre-dawn meal (suhoor) while in a state of ritual impurity (junub), and did not perform ghusl until after Fajr due to the cold weather?

The fast of one who is overtaken by dawn while in a state of janabah (ritual impurity due to sexual activity) is valid, according to the hadith of Aisha and Umm Salamah that the Prophet, peace and blessings be upon him, "would wake up in the morning in a state of janabah from intercourse, then he would perform ghusl (ritual bath) and fast." It is obligatory for a Muslim to perform ghusl to offer the Fajr prayer at its prescribed time. If one fears harm from cold water, they should warm it. If they are unable to warm it, they should perform tayammum (dry ablution). It is not permissible to delay the prayer beyond its time.