What is the fastest possible transatlantic flight today?
If you prefer something more realistic and readily available, go with the fastest business jet, the Cessna Citation X, or if you need a little more range, the Gulfstream 650. The Cessna can cruise at Mach 0.935 over maybe 3000 miles (officially 3460 nm, but certainly a little less when flown at top speed), and the G650 has a range of 6000 nm at Mach 0.925. Thanks to @Lnafziger for helping with the facts. Bonus: Both are useable from a lot more airstrips than any airliner.

The flying time from London to New York will be less than 5 hours - still quite a bit faster than flying with an airline. By the way, the airliner record on this route is still held by the Vickers VC-10 with a tad above 5 hours. If you go from London to Boston, the Cessna will have just enough range, for all other routes I recommend the Gulfstream.

The bureaucratic procedure for making an F-22 available will give your protagonist ample time to cross the Atlantic dozens of times in a business jet. Add to this the stationing of a refueling tanker in the middle of the Atlantic (hint: That alone takes more time than what the F-22 needs to fly the full distance, so its realistic mission time will easily double).