The important thing here is that spacecraft trajectory planning is complex. There's a reasonable amount of math involved in solving your problem and I suspect that the readers/viewers/players of your creation seem unlikely to be there for the mathematics. You might consider therefore conserving your details, because this is stuff that you can get wrong, and getting it right doesn't necessarily add any more to your story than handwaving in the right trajectory changes to come to a satisfyingly dramatic conclusion. Everyone is happy to suspend their disbelief in exchange for a good story, and plenty of good scifi authors fudge their figures, deliberately or accidentally.
The biggest issue you'll have is the sheer slowness of everything. Solar escape velocity at Uranus' distance from the sun is about 9.6 km/s, and Uranus is nearly 3 billion kilometers away. It'll take years. You can't be going much faster than solar escape velocity, because gravitational assists are extremely limited in the amount of change they can impart, and they have an optimal approach velocity vector and the assist is much lower if you're going too fast. If you go barelling out system at hundreds or thousands of kilometers per second (which is the sort of speed you'll want to be travelling if you aren't interested in multiple-year-travel-times) then a gravity assist will remove too little velocity from your spacecraft to keep you from shooting into interstellar space. If you're going out at 10 or 20 km/s... well. You can look at Voyager 2's planetary schedule to get an idea of how long things can take.
Take a look at this space.SE answer to give you an idea of how powerful a gravity assist can be: How much delta-v can we squeeze out of a gravitational slingshot and what factors limit it?.
The $\Delta V$ you can get is:
$$2\,v_\infty\over 1+{r\,v_\infty^2\over\mu}$$
Where $v_\infty$ is the relative velocity of your spacecraft relative to the body you're getting an assist from, $r$ is the point of closest approach, and $\mu$ is the standard gravitational parameter which is the gravitational constant $G$ multiplied by the mass of the assisting body. The linked answer contains a nice graph that shows that if you're too slow or too fast, the amount of assist you can get is limited
The very best assist Uranus can offer you is when your incoming velocity matches $\sqrt{\mu/r}$. At a cloud-skimming altitude of 26000 km, that gives you a delta-V of nearly 15 km/s. That's a lot by the standards of modern space probes, but it is peanuts when looking at how fast you need to be going to reach Uranus in less time than it takes to settle down and raise a family.
For a more detailed worked example, you could have a read of the flyby section of Orbital Mechanics and Astrodynamics. Its a fiddly process, and as such I'm not going to do it for you this time.