It'd be a pretty tone deaf political move to allow a foreign power to buy a well known UK institution given the current little-island campaigning by the government (even if they try and spin it as the idea we trade with other countries outside the EU).
Politics aside, there's often the mindset that a foreign company would look to increase "efficiencies" and then offshore elements. Not many people will cry tears over lost City jobs, but there'd be an interesting narrative at play.
The imagined scenario is economic warfare by the Chinese government forcing HKEX to just stop trading on the British stock market. Given how important the stock market is to a modern economy it'd be a powerful hit even if it only stayed off for a few days.
The current situation in Hong Kong? Selling something like the LSE, only to risk it being controlled by China in the near future should automatically result in the UK government blocking the sale even before it's consider any further.
