Most TI high performance ADC's (PCM4202, PCM1804 and PCM422x) have a hardware controlled mode. (meaning, pull up/down on pins to configure).
However, changing sample rate isn't as simple as switching one pin in many cases, as various clock dividers need to be configured.
e.g. your divider settings for the sample rate will be different if your clock source is a 24.576MHz clock vs. a 12.288MHz Clock.
In addition, your wordclock interface that sync's to other parts of your studio is a separate circuit. That will also need configuration.
Then don't forget your digital interface out... such as a S/PDIF transmitter.
What I'm trying to say, is that there are way too many variables in play in the design of a system to simply have one switch that jumps between sample rates. This is typically one of the roles of a microcontroller, to take user inputs (such as you pushing a button) and then step through states and flip multiple output pins to configure the ADC, S/PDIF and PLL circuits.
An Arduino, or an MSP430 can do this easily. Your process psuedocode will look something like this:
1. Detect user button press
2.Debounce button press (count to 1000 and see if the button is still being pressed)
3. Lookup next sample rate settings (which pins need to go high, which go low)
4. Put the ADC,S/PDIF and PLL into RESET (by holding a certain pin high/low)
5. Update your control output pins on your microcontroller.
6. Release RESET allowing each of the devices to start running again
7. Enjoy the lovely tone of these ADCs