
The PCB was made against schematic shown in post #2, so if there is a mistake in the PCB it should also appear in the schematic.I hate to say it, but we get this kind of question all of the time...there is likely a mistake in your physical circuit.
You shouldn't have gotten a complaint from moderation on this site because they prefer to not have broken links when threads are referenced in the future.Long ago, I shared a problem on two different forums. I got strong reaction from the moderators that I should have posted the link instead.
What does present mean? The IC is physically installed in the board? Or the output is HIGH?a) If TIMER2 is present

The presence of TIMER2 shouldn't have any effect on the output of DELAY2. TIMER1 triggers DELAY2 which triggers TIMER2.I have mentioned two conditions i.e. IC present and IC removed.
This means, IC is present in its base or it is removed.
Yes, TIMER2 shouldn't affect the output of DELAY2 bcz the former is activated by the latter. But physically it is being effected as there is no output of DELAY2 when TIMER2 is in circuit.The presence of TIMER2 shouldn't have any effect on the output of DELAY2. TIMER1 triggers DELAY2 which triggers TIMER2.
Other than different delays and frequency, the "2" cascade is the same as the "1". Have you verified that things are wired correctly and that there is no board defect?
10nF should be sufficient for AC coupling the trigger from DELAY2 but what was the reason that you used 100nF for the trigger to DELAY1?
Try triggering DELAY2 manually and see if you can capture its output pulse while monitoring the output of TIMER2. If you have difficulty doing that, use a timer to make an astable to keep retriggering DELAY2 so you can see why it isn't working.Yes, TIMER2 shouldn't affect the output of DELAY2 bcz the former is activated by the latter. But physically it is being effected as there is no output of DELAY2 when TIMER2 is in circuit.
With reset on PULSES2 not being driven, the outputs of DELAY1 and TIMER1 can enable the PULSES2 astable. A sufficiently low value pull-down resistor would prevent that.Moreover, I could not understand why the PULSES2 generates pulses when TIMER2 is absent.
Manual triggering DELAY2 shows output on its pin 3. But after the delay, TIMER2 does not get triggered.Try triggering DELAY2 manually and see if you can capture its output pulse while monitoring the output of TIMER2. If you have difficulty doing that, use a timer to make an astable to keep retriggering DELAY2 so you can see why it isn't working.
Yes, you have read the schematic correctly.the only path I see to ground with Timer 2 removed has an LED and a diode in series with it...if I am reading the schematic correctly.
None of the other timers can drive the reset pin high due to D3 in the upper schematic which has no descriptor in the lower one.
Monitor the falling edge of the output of DELAY2 on the other side of the AC coupling cap to verify whether that node is dropping to VCC/3.Manual triggering DELAY2 shows output on its pin 3. But after the delay, TIMER2 does not get triggered.