I had a doctor appointment yesterday and I had to wait like an hour outside.

"wait like an hour" is extremely casual speech, but correct, so I haven't changed anything but want to make a note. "Like" is usually used in spoken English as a filler sound, like "ummm" or "ahh". You might hear something such as: "He's like, really cool." However, in written English, where you are not saying things immediately, these filler sounds are usually not included. If you want to show uncertainty I would say: "I had to wait about an hour" instead.

I know that the system is overloaded, but I felt like I was loswasting my time.

I just said "like" is casual English, but this "like" means "as if", and so is a separate use case than the "like" from before.

Once I got there, the doctor was very nice and attended to me very well, h. He was not in a rush trying to attend to me quicker.