T1 (thanks for the reference designators!) is wired as a switch rather than as a linear amplifier or buffer. As such, it is "converting" the capacitor's voltage ramp into a very rapid ramp that looks like a switch to the eye.
I don't see any reason for D2, and in fact I think it hurts what you are trying to achieve. I recommend:
Eliminate D2. Connect the left side of R2 directly to pin 2.
Disconnect the T1 emitter from GND.
Disconnect the T2 base from the T1 collector.
Connect the T2 base to the T1 emitter.
This turns the T1-T2 transistors into a Darlington pair, a current amplifier that maintains the input voltage wave shape. BUT -
The circuit probably will not run well on 5 V. The capacitor voltage ramps up and down between 1.67 V and 3.33 V. 3.33 V minus two Vbe forward voltage drops does not leave much to run the LED. You might have to adjust R6 to see anything happen. A simple thing to try is to run the circuit on 9 V or 12 V.
The problem is the high impedance and relatively small voltage change across the timing capacitor. A common way around this is to add a second R-C network at the 555 output (pin 3) and connect the darlington pair to that.