Kim N. wrote:
> Is there a faster microcontroller than the ATmega2560 to solve this?

Up to Teensy 4.0 with 400MHz.

But probably your ATmega256 is fast enough if programmed efficiently.

Let me just guess:

You are using several slow uarts with the arduino framework in blocking 
mode and now you need a faster microcontroller?

For your task everything with enough hardware uarts should be suitable, 
the problem is that some programming skills are required as it was 
already mentioned above.

You have two options:

1. Learn to use a microcontroller without a kid-friendly but clumsy 
framework (or at least google for a better implementation of the uart, 
that would already fix your problem)

2. Ask how to use a raspberry pi for your application, because of your 
demand on huge processing power (to keep up with modern arduino makers)

Kim N. wrote:
> Since it's
> processing a lot of commands, there's a noticeable delay with speed
> updates in the display.

Maybe, on over 1000 different commands the parsing speed may have 
influence.
But I think, there are other time gulping tasks on your program, e.g. 
slow display speed, which can not be solved by a faster CPU.
A typical programmers fault was, that the display was written dozen 
times faster, than a human was able to read.

Which speed use the UART?

Kim N. wrote:
> Have a look into my code

Of course this is slow, because you have programmed it slowley.

All the serial.print slow down the program to the rate of the serial 
port, even a digitalRead does take 400 times as long as a 
PORTD|=(1<<BITPOS) .

Your program will not run faster if using a faster processor, it must be 
decoupled from slow I/O.

First remove ll the unneccessary serial. commands.

Then remove all the blocking while loops.

Simply told: Learn programming. Learn what's behind.