NYC still requires fingerprint checks.
In Austin, the mayor and people voted in favor of the checks, but the state legislature killed the city's fingerprinting rule. Some argue this diminished the city's ability to self-govern. That part of the history, despite being quite short, is left out of this article.
Good luck with that, the markets are not remotely comparable:
Austin: 900k pop; In a red state known for low regulatory burden; No established mass transit to compete with; No established tax companies
London: 8.8M pop; High regulatory burden; City's incentive: each Uber rider is a potential lost customer for the Underground; Taxi companies have influence with legislators
The underground is way too crowded, but so are the roads (mainly because of taxis these days).
This makes Uber a better bet than taxis (more efficient) buses better than both, bikes best of all and something in between a nirvana.
However, none of that matters as long as Uber think they're above the law.
TfL has the incentive to follow London mayoral policy; the MoL has concerns beyond travel.
But that just begs, er, fully reduces to the question of ... why haven't they? Is a slick, user-friendly app really that much of a barrier? The article has tweets that are (nominally) from users who are freaking about being able to get a safe ride home late at night, so they at least believe there aren't good substitutes.