The UK always had its own visa regime, even within the European Union so the simple answer to your question is no, it's not possible to have a single visa to visit all these countries.
Furthermore, Ireland, Cyprus, Romania, Bulgaria, and Croatia are also in the European Union but outside the Schengen area. Some (but not all) Schengen visas allow you to visit Cyprus, Romania, Bulgaria, and Crotia. Some (but not all) Cypriot, Romanian, Bulgarian, or Croat visas allow you to visit the other countries in that list. Some (but not all) British visas allow you to visit Ireland. Best case scenario, you could be able to visit the whole EU with a multiple-entry Schengen visa and the right type of British visa. Worse case scenario (in particular single entry visas), you could theoretically require 6 or 7 visas.
Brexit is likely to have an effect on EU citizens and their relatives and in particular on the visa exemptions and simplified procedures enjoyed by the members of an EU citizen's family. I would not expect any immediate effect on other visitors from outside the EU ("third-country nationals") but it's difficult to know. That's complete speculation at this point but it could paradoxically free Ireland to join the Schengen area if it wants to (as a so-called "hard Brexit" would make it very difficult to maintain the current open border regime between the countries).
Small terminological note as you are asking about that too: Great Britain is technically the main island of the United Kingdom but "Britain" or "British" are often used to designate the whole polity (as I have done above).