Like so much else surrounding Brexit, the answer to this question is unknown. However, unless and until an agreement to the contrary comes into force, on the date of the UK actually leaving the EU (in around two years' time), strictly, a UK passport holder will not have a right to stay in an EU country longer than any visa strictures allow (if a formal visa arrangement has been enacted) and certainly not to work there (again, unless an agreement to that effect has been enacted).
In practice, some or all EU countries may not actually bar any UK passport holder from entering without a visa agreement being in force but that person will (again strictly, all things being equal) then have no rules available to define his/her status. It seems unlikely (but not impossible) that some/all EU countries will regard the EU type UK passport invalid because it makes a by then invalid claim of EU status. Over time all the current UK passports will be replaced with a different one to a UK government design.