Is it permissible to silence people by saying, "O people, remember Allah," or "Send blessings upon your Prophet," when they are speaking, with the intention of gaining their attention?

There is no harm in reminding people by saying "Remember Allah" or "Send prayers upon the Prophet," as the apparent intention of the one reminding is to alert the heedless, not to perform an act of worship to Allah by that specific utterance. Such an act is permissible, as was mentioned by Sheikh Ibn Uthaymeen.