What is the ruling on teaching English to children in governmental religious institutes, and does this contradict the prohibition against speaking in the manner of non-Arabs?

There is no harm in teaching children the English language. The prohibition by thelaf of learning the gibberish of non-Arabs was for situations where there was no need. Some Companions, however, learned other languages out of necessity, such as Zayd ibn Thabit learning Syriac by the command of the Prophet, peace and blessings be upon him. The Muslims' need for English today is pressing, as it is the language of many peoples.