airplay mirroring, versus upload internet speed

You would need to make sure no other activity is occurring (other streaming, web use, network backups etc) to achieve the best results. If on wifi try connecting via ethernet. If you have a computer you can go to istumbler, netstumbler or similar to get a network report. This will show the current status of the network (signal strength, noise, nearby networks).

Hi. Mirroring only uses your local network, so it's not affected by speed of the internet connection. If your network is lightly loaded, the biggest influence on mirroring performance is the graphics processor in the computer, since it has to convert the screen frame buffer to an H.264 video stream for transmission.

As I said, mirroring uses the computer to convert the screen image to a real-time video stream for transmission to Apple TV. So the computer is busy playing the video, and converting the screen to a new video for transmission. When you use direct streaming, the original video is simply transmitted to Apple TV. This also affects your network traffic, since there is only one stream with direct streaming, but three streams with mirroring: incoming video to computer, mirror video to router, and mirror video from router to Apple TV. If your network has other traffic, the performance may be affected.

Are you using the same computer at the other home?