Apostrophes in English mean one of two things:
1) Possession or association, which you now understand
"London's finest" without anything else is arguably incorrect because it's lacking the subject, but would usually be interpreted in context (e.g. if you're talking about parks, it'll be London's finest parks).
2) Deliberately missing letters, e.g. can't is actually can not - the apostrophe indicates that letters (in this case, a space,n and o) are deliberately missing. This usage is more complicated because it's based on common usage.
Adding an 's' to a word without an apostrophe makes it plural (although, of course, not all words are pluralised that way). So "Londons" means "more than one London" and in that context "finest" doesn't make much sense. It could be interpreted as "the finest <whatever the context is> in all of the places called "London" in the world and I'm not sure about how to show possession with a plural that ends in s".
"London's Finest". If the slogan is just that, then it's claiming that either the T-shirt or the person wearing it is the best in or from London, or at least amongst the best grouping in or from London. Is that your intended meaning?