After reading the source code, RH_ASK.cpp, I found a commented out macro.
// RH_ASK on ATtiny8x uses Timer 0 to generate interrupts 8 times per
// bit interval. Timer 0 is used by Arduino platform for millis()/micros()
// which is used by delay()
// Uncomment the define RH_ASK_ATTINY_USE_TIMER1 bellow, if you want to use
// Timer 1 instead of Timer 0 on ATtiny
// Timer 1 is also used by some other libraries, e.g. Servo. Alway check
// usage of Timer 1 before enabling this.
// Should be moved to header file
//#define RH_ASK_ATTINY_USE_TIMER1
In other words, the default behaviour of RadioHead ASK init() is to use Timer 0, which causes delay() (which uses millis()/micro()) to behave incorrectly.
To fix: Tell RadioHead ASK to use Timer 1. Because the RH_ASK_ATTINY_USE_TIMER1 macro is in the source and not the header (for now), you have to modify the actual RadioHead ASK source file, RH_ASK.cpp*. Un-comment the line #define RH_ASK_ATTINY_USE_TIMER1 which will use Timer 1 instead of Timer 0. Re-compile your code, and delay() should start working properly again.
* RH_ASK.cpp can be found in %USERPROFILE%\Documents\Arduino\libraries\RadioHead on Windows, if you installed RadioHead via the Arduino IDE library manager.