Can you try a clean boot for me real quick to see if it could be a program running in the background causing the timeout?
Once your PC loads up immediately right-click on Apex Legends from your desktop and select "Troubleshoot Compatibility." Then right-click on the the shortcut and run it as administrator.
Let us know if that helps.
I would also mention that I suspect it MIGHT be related to the "two people on the same LAN can't play together" issue, but really don't know if that's possible:
My IP is dynamic. My ISP rotates the same IPs through several people around here it seems. I got an IP ban on a site for stuff I didn't do and dissapears as the next IP is assigned to me. Sometimes that IP comes back to me and get banned again. Also got range bans from the same site. I think my ISP treats all clients as a single big * LAN first, and if there's anyone trying to play AL right now, it's gonna crash someone else. As I said, seems far fetched and maybe I'm talking stuff I don't know about, but just wanted to have it out there.