Typical culture wars approach - failing to even consider the political issues behind changes in crime levels. Ideally, let's slip off at a tangent and say wow to some celebrity rape charge.
It is worth asking why it matters and to whom if London has better or worse crime statistics than New York. Is this like comparing teams in a league table? And of course, while the trends may be in opposing directions, it does not follow that actual levels of crime have yet equalised.
Does someone have the idea that London, the heart of British capitalism and the engine room of Tory Britain's neoliberal project, is a socialist dystopia to be contrasted with good old free market New York, hardly the most radically right wing of American states?
Crime is an output measure, it is an indicator of social health, not an isolated phenomenon. Crime is driven by social inequality. So is health, so is educational attainment, so is every other useful measure of human well being.
Crime is also a measure of the way crime is defined and measured. The City of London is the corrupt heart of the world's greatest financial services based criminal empire, incorporating a global network of tax havens on British controlled islands from Jersey to the Cayman Islands, not omitting Hong Kong for all the transition back to Chinese government. The big money in British politics is with the City of London's hedge funds, not least in funding the Brexit campaign and the UKIP Party, as well as Nigel Farage's hectic stirring of fascist movements across Europe and of course Trump's America. Still, that will not feature in any discussion of "crime."
To the extent that crime is influenced by policing, it is relevant that the neoliberal Tories have slashed spending on the police to the extent that many forces no longer investigate minor crime -they just don't bother with it. That does not prevent them increasing spending on a prison service that is increasingly privatised, but remember the American Heritage Foundation has its hand directly in the British cabinet.
If you want to understand crime levels, look at evidence of grotesque and escalating inequality.