Do UK authorities have access to data about the arrival and departure of people within the Schengen area?
Not even Schengen nations have access to these data! That's why they stamp on entry and exit. Each Schengen nation only has electronic records about entries and exits from Schengen via its own borders. Otherwise, they have to look at your stamps.
There is provision to share when visas are issued (via the Visa Information System), but the UK does not participate. And there is provision to share some information regarding entry refusals and deportations via the Schengen Information System. However, the UK only participates in sharing of information regarding wanted criminals and missing persons.