555 timer to drive transistors with low load resistance
Your filament has a 'lit up' resistance of 2V/1.5A = 1.3&ohm;, but a cold resistance of only 0.2&ohm;. If powered by a constant 2V it would initially draw 2V/0.2&ohm; = 10A. So for fast 'lighting up' your circuit should be able to provide at least 10A at close to 2V, while not exceeding 2V when the filament heats up.


  That's why I tried to put another BJT tried to increase current(maybe
  it's a bad idea).


Your idea is good. The MJE3055 can (just) deliver 10A, but only has a current gain of ~15 at this Collector current, so it needs a Base current drive of ~10/15 = 0.7A. The 555 Timer IC can only source ~0.2A, so you do need the extra transistor to amplify the current.

You main problem is that R5 and R6 are limiting Q5's Base current to ~0.5A, so it is not capable of providing the required 10A output. Replacing R6 with a short will  increase Base current to ~0.8A which may be enough. If that is still not enough then you can reduce the value of R5 until it is.  

But then you have another problem. When the filament heats up and its resistance increases the voltage will rise above 2V, which will probably burn it out. You need something to regulate the voltage, so it stays relatively constant even as the filament current varies. In the circuit below this is achieved with an extra transistor acting as a voltage comparator, and some diodes providing a voltage reference. 







simulate this circuit &ndash; Schematic created using CircuitLab

Diodes D1 and D2 drop ~0.6V each, providing a reference of 1.2V. Q3 compares that voltage to the output voltage, and begins to turn on when it gets ~0.6V higher (ie. at ~1.8V). It then shunts some of the current coming out of R4 to ground, reducing current supplied to Q1. As output voltage rises more Q3 turns on more and shunts more current away from Q1. In this way it regulates the output voltage to around 1.8~2.0V.       

Notes: 

R2 puts ~10mA through the diodes to ensure that they drop the required voltage even when Q3 is off. 

R1 limits Q3's Base current in case a fault condition causes the output voltage to rise above 2V (eg. filament blown and/or Q2 shorted out). 

I increased the value of R4 to reduce power consumption in Q3 and the 555.