Your transistors are emitterr-followers so the resistors are not needed and cause the output voltage to be very low.
Emitter-followers waste 0.7V to 0.9V for each transistor and your circuit with many transistors has two transistors in series wasting about 1.5V to 1.7V. You should change the transistors to be common-emitter switches then they will have a low voltage loss.
I expected a voltage loss of around 2V, and in this application, it might be needed.
Why do you have a positive power supply and a negative power supply? Why not use just a positive power supply? How many volts?
I am running this from a 12V 8Ah SLA battery. those symbols were just handy. I'm not that good at drawing schematics.
What is the forward voltage range of your LEDs (typical voltage and maximum voltage)? What is the current their brightness is rated at? What is their maximum continuous current?
LEDs are typical 1W buttons on star heat sinks. Vf of 3.2-3.8. Running without current limiters, so they are getting whatever is able to be passed through the transistors. Max current at 0.1% duty cycle is 1000mA (1A). Max sustained current is ~350mA.
You have the LEDs turned on for only 2.3ms. Durations less than 30ms appear to be dimmed.
Running 3 LEDs in series for a V drop of 9.6-11.4, with the extreme short on time is giving me a very bright flash, able to be seen for about 500 meters, tested outside on an overcast day.
The reason I am wanting to use the extra transistors is to lower the load seen by the 555. I tried using several transistors directly from the output, but that warmed up the timer more than what I am comfortable with.
I've used the MPSW01As because that happens to be what I have on hand that will handle more than 500mA. I do have a few P55NF06L power mosfets (able to handle 50A at 60V), but I'm thinking those will pass too much current.
I'm probably wrong about this, but I was thinking that by controlling the base current, I would also be controlling the current through the C-E connection.
If I need to. I can feed the LEDs through an LM317 set up as a 500mA current source, but that may also limit the brightness of the flash.
Right now, I'm just experimenting around with this, before admitting the possibility that I'll have to buy xenon flash to achieve what I want.