The short answer is :
You could. But GA traffic is heavily discouraged both in terms of fees and policies.
Most of the time, the only private jets you'll find at Heathrow are those operated by heads of state and their respective governments who, for whatever reason, feel they absolutely must land at Heathrow.
Most private private jet traffic that needs to be as close as humanly possible to Central London goes to RAF Northolt. A military airfield that's (pretty much) the same distance to Central London as Heathrow. But its far less busy than Heathrow (and probably cheaper too, I don't know the prices off the top of my head).
The added benefit of Northolt is that security is (obviously) MUCH higher than Heathrow. Which makes the job of keeping your VIPs safe (and the paparazzi and unwashed masses away from them) much easier.
Military needs obviously take priority at RAF Northolt in terms of availability, but most of the time its not a problem. The RAF do most of their stuff from bases outside London out of respect for the NIMBYs.