There are some mud clients (I can't remember the one I used on Windows, but it did this) which parses out a lot of information and lets you have a multipane/window view of the text. This can help. It's also possible to turn many of the notifications off. Also, you usually start in something like a common room/town and those tend to be
very active with people moving in and out constantly or carrying on non-shouted conversations. Once you get out of those areas the speed of text tends to drop off. Finally, with room descriptions you can always "look" again. I became very good at reading 1-3 lines of text before it disappeared, pulling it up, and starting right where I left off. But honestly that didn't happen too often.
In combat, depending on the MUD and the class of character you may not need to type frequently during combat. Usually you will attack automatically once it begins. Spells, healing commands, and other things are all you need to type and usually not every round (which is about 1-3 seconds usually). Every MUD I played on had some form of alias, and if not the client did, which let me take a 5 word command down to 1-5 letters. `draw crysknife` became `dc` for my Fremen character. I had similar aliases for putting it away and drawing other weapons or switching to other attack strategies.