We need to be a bit careful to differentiate between the need for a Visa and a Passport.
There are well over 100 countries where UK passport holders usually require no visa to undertake a visit either for business or pleasure. Among perhaps the more surprising nations for which no visa is required are Argentina, Taiwan, South Korea and Uruguay.
As far as I can recall (though it’s going back a bit) no visas were required to visit the western European nation s pre-EU, but I’m not sure about the more recent additions in the east.
The “borderless” arrangement for members of the Schengen agreement will not alter as the UK is not a member of that group. At present UK passport holders entering any of the Schengen nations from the UK need to produce a passport, but thereafter they can travel between other member states without hindrance.