Unfortunately the info in the listing you got those from is pretty much nonsense.
(You can run some LEDs directly from a lithium coin cell, but only because the cell cannot provide much current - somewhere from three to ten milliamps, probably).
True UV LEDs generally have a minimum forward voltage somewhere in the 4 to 6 volt range; if they emit anything at all at 3V it will be a fraction of their actual rating.
To give constant light output, LEDs need constant current, which means either a resistor from a fixed voltage or an electronic current regulator circuit, if the voltage difference between the LED and supply voltage varies significantly.
eg. if the LED did work at 3.3V as in your sketch, the brightness would change by over 4:1 between the battery being fully charged at 4.3V an near dead at 3.5V, as the voltage across its series resistor changed from 0.9V to 0.2V
For the solar cell, the simplest I'd consider is a singe schottky diode in series to the lithium cell, plus a close tolerance 4.1 or 4.2V zener directly across the lithium cell to limit the voltage.
The cell peak power point is 4.46V at 5.9mA (in full, direct sunlight).
Give more information about the size and weight limit for the project? That allows people to suggest possible solutions.
For info, any UV LED may be hazardous if people can look directly in to it, or it's illuminating the same area of skin for long periods; the light should only ever be indirect and only from UV-A wavelength LEDs. UV-B or UV-C should not be viewable at all.
The LEDs you show are not likely to be very powerful, if they even are genuine UV - but it you are selling anything you need to be aware of any possible hazard.
This is a typical warning label for equipment with UV-A LEDs: