The problem is that some data can only be shared when sufficient legal frameworks are in place. Don't expect data from an EU database containing data on EU citizens to be shared with a country that doesn't adhere to minimum data protection standards approved by the EU.
Fortunately the UK will keep GDPR so some of that should be covered. Access to SIS II (the Schengen Information System) is less certain as that requires very clear agreements before that data is shared. For instance, Denmark is an EU member state with a higher status than the UK will be post-Brexit and even they are not allowed real-time access (useful for controlling borders for instance*) because they have opted out of the justice cooperation package that would ensure this data is treated correctly.
* SIS II contains, among other things, details of people under requests for extradition; missing persons; requests by a judicial authority or suspects of crime and data on lost or stolen firearms, identity documents, motor vehicles and banknotes. Quite useful information if someone rocks up at your border travelling on a Swedish passport and claiming to be Per Johansson from Jönköping and here for a city-break.
Biting the hand that feeds IT
