Well, to be frank, I would not say, "living in London". It does not sound right to me. I would say "I need to visit my mother in London".
If you think you need to confirm to someone that your mother lives in London, and you need to go there to visit her, then I would say, "I need to visit my mother who is living in London".
The comma is used to set off a clause or a list of adjectives or adverbs. "I need to visit my mother living in London "., is a single clause " is a single clause and has no series of adverbs or adjectives that need to be set off with commas.
This is my opinion. Let's see what others may have to say.