The journey proposed in the OP, Los Angeles to London, obviates much of the talk re: private jets. The only private business jet that has the range without refueling would be the Gulfstream V. Several other jets are marginally faster overall, but don’t have the range. Note that I’m not including the Boeing Business Jet in this category, as you can get virtually any of Boeing’s planes in a business jet version.
Since the route would be over uninhabited areas for much of the journey, a supersonic aircraft would seem ideal; however any privately available supersonic plane, such as a former military fighter, would need to refuel at least once in the journey. So our best options are a commercial 747-400ER flight, renting a private 747-400ER, or renting a Gulfstream V.
A commercial flight is the least fast of the options, because of both the increased load, and the tendency to fly at max economy, rather than max speed. Commercial flights are listed as taking 10 hours, 30 minutes.
Between the private 747-400ER and the Gulfstream V, the 747-400ER is faster, and the 747-400ER has the edge in range; particularly as you would likely have a much reduced load - potentially just you and your luggage. If you take even 12 people, the loading of the Gulfstream becomes problematic for speed vs. range, while 12 people is a negligible load for the 747.
In any case, a private flight will be quicker than a commercial flight, for sure, since the maximum speed of at 747-400ER is about 100 knots above max economy speed. That potentially shaves 15-20% off our time - so figure 9 hours for the trip.
Since we are talking unlimited funds, assuming my earlier time travel suggestion unrealistic, I suppose one could develop a long-range supersonic jet, to further reduce the time required by maybe 25%. LAX to LHR is too far for either the Concorde or the Tupolev Tu-144 to make without refueling. One could also buy in-flight refueling for ones private military jet, come to think of it. So times under 8 hours are possible, even without any exceptions to supersonic transport over populated areas.