1 meg is a little high, I would drop that to 33k or 47k, and reduce the 10k pulldown to 1k. Then change the value of C1 to give the desired time constant.
Also you labeled C1 as 220 Farads!
Thanks for pointing out the Cap error. I have replaced the attachment with the correction.
I don't see why you recommend reducing the resistors.
In fact I used 10M in a similar circuit in the Garage Door Opener that I designed some time ago.
It has been working for several years without any problem.
I used it to keep the light on for about 3 minutes after the door has been opened or closed. So the timing is not critical.
As for the 10k: many years ago, the rule of thumb for switch & relay contacts was that they need at least 5 Volt across them when open & at least 1 mA through them when closed.
I don't know if this applies to modern contacts, but I abide by it anyway.
So the circuit does not meet this criterion for a supply voltage of < 10 Volt so the 10k could be reduced in proportion.
Modern practice is to use much lower currents through the switches.
I have a problem with my Printer/Scanner as the Scan button often does not work, so I have to press it 2 or more times.
So I wonder if that issue would not occur if they had obeyed the "rule of thumb".