How? There is a well established separate court system in Scotland so I think that part would be fine. But pretty much every other part of government would have to be built from scratch. A foreign office, a home office, immigration, passport office, etc. etc. etc.
Most of these are institutions are based physically in rUK so it’s not like they can be “split”
How could it be judged that institutions are stable immediately after independence? There'd be a ton of new ones that would need to be set up.