Privately owned by a man named Paul Buckland
According to a 2005 BBC article, it was sold at auction in the 80s:
So, what happened to the car? Well, it wasn't ever the property of the Beeb - the car was always Jon's. After the show it ended up on a trailer and was becoming a bit of a burden. In the end it was put in a car auction in the early 1980s and was bought - for £1700 - by someone who was purchasing the car for his son. The car is now in a secure location in the UK and hasn't seen the light of day since 1996.
One of the car's rare appearances was in the 1993 documentary 30 Years in the TARDIS; Buckland was thanked in the credits for providing the vehicle:
It's most recent appearance was at a 2015 model show in Hanslope, England, apparently still in the hands of Buckland (later images on the page identify him as the gentleman in the grey shirt):
In a video of the car's unveiling, posted to Facebook by an attendee, show organizer Mat Irvine1 introduced the current owners as Paul Buckland and another man (whose name I can't make out from the video).
I've read it claimed that Pertwee gifted the car to a fan at some point after he left the show, but I've been unable to find a more official source for that.
1 As an interesting aside, Irvine was a visual effects designer for the BBC for a number of years, and built several of the K-9 props (including the original one used in "The Invisible Enemy")