Alternatives to RC circuit for simple variable delay up to 2 minutes
Depending on how you measure "simple", a small microcontroller is a good answer.  Even the tiny PIC 10F200 can do this easily, and comes in a SOT-23 package.  That's the same package single transistors come in, although it has 6 leads instead of 3.  The only other part you need is the bypass cap.  This circuit will be physically smaller than the one you show, and in volumes probably not much different in price considering the large cap yours requires.

The 10F200 comes with a internal oscillator good to a few percent, which is better than a cap you can get that wouldn't cost more than this micro.  This PIC can also have internal passive pullups on some inputs, so the switch connects directly between a pin and ground without any other parts required.

The delay is done digitally by counting cycles.  Delays done that way can be arbitrarily long, and have the accuracy of the internal oscillator.

Depending on what you are trying to drive, you might be able to do so directly from a output pin.  Otherwise, you still need a transistor with base resistor, or a "logic level" FET with no other parts.

One advantage of using digital logic for the delay is that the transistions will be crisp and essentially instantaneous.  In your example, the output transistor will fade out slowly.