This isn't an overnight build.
Phase 1, design
Someone has to design the craft to withstand the forces your engine provides. Otherwise, your pilot takes it out for the first test-flight, hits the accelerator, and either becomes part of the engine or watches as the engine rips forward through the fuselage and leaves him and the debris behind.
So your engineers needs to get busy with AutoCad (or equivalent design systems). Then send their rough designs over to be tested in simulations; some digital, some physical. Simultaneous to that, some group will be building various sizes and designs of this new engine, attaching it to rocket sleds, and firing it up to accurately measure it's acceleration and other forces.
Phase 2, tests
After a design is approved, various unmanned miniatures are going to be built, wind-tunnel tested, safety-tested, and then test-piloted. Each of these tests will consume time and materials. Each will result in new data that goes back into the design, so each new test is a refinement on the previous designs. It won't be perfect the first few designs. Heck, the first mock-up would probably just explode on the runway if you went straight from "back of envelope" to manned prototype.
Phase 3, test flights
At some point, you'll feel confident the system is safe enough to stick a person in. You'll find a volunteer test pilot. They will fly a prototype that's (we hope) got a working design. The pilot will discover any lingering design flaws. There may be a few changes and re-tests.
Phase 4, FLIGHT!
A production prototype will go out, for even more test flights. At this point, your design is solid. This is more fine-tuning and discovering the limits. The final build will have some build-up phases as raw materials are converted into finished parts on spec, then assembled into a working system.
Real World Examples
According to this post, it took almost 9 years for Boeing to build the 787. That's an iterative update to jet planes, so no new physics involved. Your new engine will probably have a slightly higher learning curve in the early stages than this. But the larger budget will reduce your time.
Elon Musk began hiring staff for SpaceX in 2002. The goal was to get to Mars in 10-20 years. As of 2017, they have not yet achieved that goal. This timeline is closer to your OP, since Elon Musk comes closer to "Tony Stark" than most real-world peope (at least with regards to available cash). Your question assumes more money and more resources, so the time gap will be lowered. More engineers and a harder schedule is possible, but only to a point.
The US Space Shuttle program chose a design in 1972 after starting up the project in 1969. The first flight was in 1977. The first orbital test was 1981. In 1982, the first non-test flight took place. Thirteen years from the formation of the task group to the first non-test flight. Your group will move faster, thanks to CAD, availability of supercomputers for rapid initial testing, and because budget is no constraint on the project.
Engineering for a project like this always falls back on the project management triangle. You have 3 inputs: Time, Quality, and Cost. The old maxim is "Cheap, fast, or good; you can pick any two." You want Fast and Good. So your cost is going to be quite high.
I imagine this project could not be safely completed in less than five to ten years. But some of that preliminary design work would (we hope) be happening while Mr. Stark was off in his bourbon-fueled manic design state. Hopefully, while he's tweaking the engine design, his engineers would also be building the boring old plane to carry that engine.