The established nature of the market may be a mixed bag for Square. There is strong competition, but there's also a very large market. Square don't have to convince anyone of the merits of mobile payment, just offer a better solution for some portion of the market. They already have lower transaction fees than Paypal and iZettle for most users; they could gain significant ground by salami-slicing the market with better features for particular niches.
I am really curious what was the thinking behind Square's UK launch (if there was). At least as a business traveler (and admittedly in London, the financial center of Europe), I saw no space in the market for Square to enter.
[1] (The ONLY place I could not use Android Pay was Apple Store)