My mother has a cough with phlegm, but she doesn't understand to spit out it out and keeps swallowing it. Any suggestions?

I think you are obsessing over this without reason, contrary to folk wisdom it is totally harmless for her to swallow the phlegm. Your constant hovering whenever she coughs must be irritating for both of you, I would worry more about treating the cause for it.

Try a tall glass of hot water lemon and honey and keep plenty of tissues available - none of these ingredients should conflict with her beliefs - I understand the balancing act you are trying to walk.

When my mom is anxious or agitated I find she calms down a bit if we say the Lord's Prayer together - whenever I was sick as a child she would sit on my bed and comfort me and so I try to do the same for her now.

Think of how you would get it out of her mouth if she were a baby by suctioning her mouth. Try saline in her nose and suction her nose. Try to thin it by using a dehumidifier or steamy shower. Let her drink a hot tea or cup of thin broth or soup. Stay away from milk, cheese or any dairy. Sometimes you can tap her back to help loosen the mucus. I know this is stressful for both of you.