Assalamu Alaykum Ustadh. During the Jumu‘ah khutbah, speaking is strongly discouraged. Would saying phrases like “SubhanAllah” or “MashaAllah” loudly be permissible, or would it fall under prohibited speech since it distract others?

Don't disturb others by saying it loud. I understand that what you mean is you actually vocalize it rather than saying it in your heart; that is what you mean by "loudly." It is better not to do so. However, some of the people of ilm (some of the scholars) have said it is allowed.