What is the ruling on praying at home if the nearby mosques do not allow sufficient time between the Adhan and Iqamah, leading to missing the first rak'ahs or the entire prayer?

You should be keen to pray in the mosque because of its great reward and merit. To catch the congregational prayer, you must prepare for the prayer well before the iqama. You catch the congregational prayer if you join the Imam before the tasleem, according to the majority of scholars. If you pray in congregation at home with your family, there is no sin upon you.