Presumably the UK authorities have no power whatsoever to compel Zuckerberg to do anything, but as others have pointed out, he does have considerable business interests here, on which hitherto he has paid little or no tax on. So in theory his business interests could attract the interest of HMRC, the SFO and other bodies at the behest of HMG, if they so choose.
Given he's invested in offices and at least one data centre in the UK and as such has registered business interests here it could, as others have said, be indicative of very poor business acumen - with an attendant loss of confidence in the markets (i.e. Faceberk shares get dumped as the price drops, sending the value even lower) - to extend his middle digit and invite the Select Committee to take a swivel.
He could, however, dodge the bullet without seeming to dodge the bullet by sending whomever bosses his UK operation in his stead.
Quite frankly, I'd welcome the arrogant cockstand chinning off the government to the point of angry retaliation as it would be hilarious to see every department from HMR&C to the HSE and Trading Standards tear his UK-side interests to shreds in microscopic audit after audit after audit.
The fallout could pique Brussels' interest and every business interest and asset he's got in every EU state could feel the discomfort of an administrative anal probe.
Couldn't give a rats if FB lives or dies. It is a platform for people who post pics of their breakfast and cute kittens. And their revolting children. Of more concern is the links between FB, Cambridge Analytica and Russia.
It seems that the alleged murderer refused to give over his password. That's not Facebook's fault.
To avoid such problems the Police want instant access to everyone's Facebook account and presumably every online presence they have (including this one).
Your bank details and a DNA sample to be filed for future reference too perhaps. After all everyone has something to hide and this information will be useful for tracking down non-child murderers as well.