Is it permissible for me to prevent my ten-year-old son from accompanying me to Fajr prayer due to the severe cold?

You are not obligated to take your child to the mosque if you find it difficult. However, if you are sure he will not fall ill, it is better to take him so that he gets accustomed to praying in the mosque, which will lead to his preserving the prayer. Sheikh Ibn 'Uthaymeen mentioned that regarding waking children for Fajr prayer and taking them to the mosque, one should consider the circumstances: if the weather is cold and difficult, there is no harm in letting him stay and guiding him to pray when he wakes up. However, if the weather is moderate and there is no harm to him, then wake him up so he gets accustomed to praying with the congregation. The Permanent Committee for Issuing Fatwas stated that bringing a discerning child to the mosque is legislated so that he becomes accustomed to prayer, based on the Prophet's (peace and blessings be upon him) saying: "Command your children to pray when they are seven years old, and beat them for it when they are ten."