LEDs in series but only one glowing
If the LEDs are fully within the UV spectrum, you won't see any visible light emitted from them, except for cases where the LED is being overdriven with too much current, etc.

If these are "blacklight" LEDs then they're a deep purple with some UV, and you should be able to see visible light from them.

Assuming that these are typical 20mA If LEDs with a Vf of ~3.2 volts, you could try connecting just one of them in series with a 90&Omega; resistor (100&Omega; is the closest common value). Do this for all of your LEDs to ensure that they work.

Once you've established that they are all functional, you can go back to your series-parallel circuit. However three LEDs in series will require 3 \$\cdot\$ 3.2V = 9.6V. This means you'll need to provide at least 9.6V prior to selecting current-limiting resistors for each branch.

If you want to use 5V, consider connecting the LEDs in parallel such that they each have a 90-100&Omega; current-limiting resistor.

Remove R1, the 150&Omega; resistor from the battery to ground; it serves no purpose.