You'll most likely be checked going from Britain into France, The Netherlands, or wherever you first arrive in Continental Europe. If you take your car into France by the Channel Tunnel, for example, you'll be checked by a French official before you actually board the train in England!
Once on the Continent though, there a lot of border posts between EU countries which are now almost ghost towns, and the chances are you'll just drive through without anybody even seeing you.
By the way, I got a good telling-off by a grouchy passport inspector some years ago when arriving back in London from the U.S. and I went through the wrong line.
That was just after the relaxation of the European borders with the "Single Market" in 1992. The signs had all been changed and there were just two sections marked for "EU Passport holders" and "All other passport holders." I was still on my old traditional-style British passport at that time, not a new European one, so I went to the "All other" line. Apparently we were just supposed to know that "EU Passports" actually means "EU and British passports."