kipper2k, Yes, you have that NAND stuff OK.
I like to just keep NAND gates in stock and use them exclusively for oscillators and buffers. You can get dedicated inverters that have 6 inverters to a package, but I find the layout a bit crowded.
You could also tie one input of a NAND gate high, so the output would only go low only if the second input became high, which it does eventually after the capacitor charges up enough.
Anyway, yes, you can change either the R or the C to get longer times.
You just multiply the value of R (in Megohms) by the value of C (in μF). to get the approximate time.
eg if you wanted a 5 second delay, you could use a 10 μF Tantalum capacitor and a 470 K resistor.
Don't use electros for timing. They have too much leakage current.
You can leave the 22 k resistor unchanged as this only affects the discharge time.
The diode should be a silicon type. I measured the reverse resistance of some Schottky diodes and it was around 1 Megohm and very sensitive to temperature, so these probably would not be suitable.
CMOS logic works OK up to about 15 volts, so look for the 74C... number.
These are becoming rare now as the 74HC.. devices (which have to have a 5 volt supply) take over.
These are great, but not as flexible as the 74C devices.