Recommendations For High Capacity Microcontrollers
In a comment, Juraj wrote:

the ATmega MCUs are similar so a port to Arduino Mega should be
possible

This seems indeed to be very much the case.
Looking at the source code of the library, it appears it uses
timer 2 for generating the PWM signal and timer 1 for
scheduling the samples. These timers are almost identical between the
ATmega328P of the Uno and the ATmega2560 used in the Arduino
Mega 2560. The only relevant difference between these boards, as
far as the library is concerned, is the output pin, which is the “OC2B”
pin of the microcontroller:

on the Uno, OC2B is digital pin 3
on the Mega, OC2B is digital pin 9

So my guess is that the library should work on the Mega with just this
tiny change:
--- a/Talkie/talkie.cpp
+++ b/Talkie/talkie.cpp
@@ -71,7 +71,7 @@ void Talkie::say(uint8_t* addr) {
        // 
        // Enable the speech system whenever say() is called.
        
-       pinMode(3,OUTPUT);
+       pinMode(9,OUTPUT);
        // Timer 2 set up as a 62500Hz PWM.
        //
        // The PWM 'buzz' is well above human hearing range and is

Also, make sure you plug your headphones on pin 9 rather than
pin 3.
If you try this and it works, please report that either here or directly
to the author of the library. It should then be easy to modify the
library to support both boards out of the box.

Edit: If you download Talkie through the Arduino library manager,
you will get this version, which does supports the Arduino
Mega 2560.