Driving two parallel LEDs with PWM brightness control and a "LED selection" signal
As demanded, I am giving more details on the simulation:

Because my CAD software (Kicad) is not very permisive with simulations, I recreated the schematic on Tina. The LEDs are inactive when filled in Grey, and active when blank.

Edit: I replaced the PWM N-Channel MOSFET (Q1) by a P-Channel MOSFET with the G pin linked to VCC instead of ground

It solved the PWM switching issues, now, the LED are properly switched On and Off depending on the PWM frequency and duty cycle.


New results are as shown (from simulation):

With the "LED selection" switch open (LED 2 powered), I get the following results: 

And with the switch closed (LED 1 powered), I get the following: 


Since the LED I want to use have a typical forward voltage of 3.1V at the desired current, it should work with this circuit, but the margins will be low so I have to be sure that my supply voltage is not lower than 3.7V. 
In The case where LED1 is powered, no voltage is passing through the second LED and the voltage is not high enough (2.3V in the simulation) to compensate for the foward voltage  so it will definitely not be lit when it should not be.

Note: I still have to corrige the series resistance of each LED to get the correct value of current flowing through the LEDs (approx 300mA &lt;=> 1.8 ohm for the white LED, and I still have to chose my UV LED). The values displayed here are generic.