The main problem with your circuit is that the time constant of 330 ohms and 1\$\mu\$F is only \$330\mu sec\$, which is not all that long, and the LED will not be especially bright for that brief time. You have to consider the physiological response of the human eye.
Your eye acts as a kind of integrator over a period in the 100msec range, so a very bright pulse of light for a short time (such as \$330\mu sec\$) would actually be visible, but it would have to be about 300 times brighter than a continuously-on light to achieve the same apparent brightness. So, an LED that is acceptably bright at 2mA would need 600mA pulse for ~300\$\mu sec\$, or to have a similar chunk of \$ current \cdot time \$ metered out to it. Since that has to come from the inverter output, it's a bit much to ask.
You could use much higher value resistors (such as 300K) and feed that to another gate, using the output of that gate to drive the LED.
As an alternative, this would be a great application for a dual monostable multivibrator such as a 74HC123. The complexity is not very different (4 resistors, 2 capacitors, 1 chip and no diodes). It's a bit different because this circuit does not stretch the existing pulse, rather it produces a visible chunk of light on every valid edge (either positive or negative, depending how you wire it) of the input signal.
Drive the /A or B input of each multivibrator with the BT signal and tie the other one inactive. (For example, if you want it to trigger on the falling edge, use the /A input and tie the B input high). The reset input /R is active low so it should be be tied high.
You can drive the LEDs by connecting them to Vcc through a suitable current-limiting resistor (such as 330 \$ \Omega\$ ) from the /Q outputs.
The time constant from Rx and Cx should result in a pulse that is easily visible, so somewhere in the 200msec range would be good. For the TI part, the time is
\$ t_W = R_X \cdot C_X \$, so 470K and 1uF would be reasonable.
The 1uF capacitor has to supply only microamperes to keep the monostable in action, and the monstable output does the heavy lifting- providing ~10mA for ~200msec to the LED.