Is Master Program Rank Important for PhD Application
I don't think it's so important for terminal masters degree, although it should not be completely disregarded. What is much more important, in my opinion, is the exact field you are intending to work in and the person you will work with and the references that you will get in the end (and, again, who they come from).

So if Edinburgh would give you the chance to work with a leading scholar in a subfield you are interested in, and UCL is ranked 10 ranks higher generally, but not as compelling for your more specific interest, I would go with Edinburgh.

Also, 10 is really not much of a difference.