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Currently, I'm learning about the UKF and in order to understand it in a good way, I programmed it, however in order to see it working I need a problem to solve which I can't now find.

Can you please recommend a very basic problem in which all the equations and the data are explained and ready to implement in my code?

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I haven't tried it, but the IEEE Proceedings paper by Julier and Uhlmann in section VI. B has a nice example:

Consider the problem that is illustrated in Fig. 8. A vehicle enters the atmosphere at high altitude and at a very high speed. The position of the body is tracked by a radar which measures range and bearing. This type of problem has been identified by a number of authors 1, [9]–[11] as being particularly stressful for filters and trackers because of the strong nonlinearities exhibited by the forces which act on the vehicle.

Figure 8 of the paper.

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