It seems that the developers have reverted to an old version. This may fix the issue, although I still urge bezbro to release a new version that can pick between P2P, Relay, and a direct IP Connect. (Assuming connection was the issue here, although even if it wasn't it's still a nice future proof)

First, you must open Steam's built in console. This can usually be done by opening
in your web browser. This should result in a new tab opening on steam next to your username saying "Console". Click it and you should see a mostly empy screen with a console and a box for input at the bottom.

The recent update switched from P2P to Relay. This seems to have broken multiplayer by resulting in player's position movement not being synced. We're going to downgrade to the version that used P2P.

You should be in steam's console right now.
At the bottom on the screen, click on the input box and enter

This will display a message along the lines of:

Now you must wait, there is no progress indicator, just wait. You'll eventually see a new message in console telling you that the download finish and where it's located.
For most users, it'll be located in:

Open file explorer and navigate to this directory.

You now have the depot downloaded. Now you must launch labrat!

Open file explorer and navigate to the directory where the depot saved to (see previous section, usually "C:\Program Files (x86)\Steam\steamapps\content\app_1402020\depot_1402021").

You should see the files for labrat. Run the game.
No extra setup is needed, create a lobby and have your friends join with the direct connect mode.
(Remember, sometimes labrat will have connection issues, if it doesn't work:

1. Try having the user that cannot connect: restart the game
   
2. Try having the user that cannot connect: restart their pc
   
3. Have the server host create a new lobby WITH A DIFFERENT NAME AND PIN

).

Bezbro, instead of forcing us to use a specific connection option, PLEASE:
Allow us to pick between, P2P, Relay, or just directly connecting to the IP of a dedicated server.
This'll future proof the game against bugs and when you inevitably shut off the servers in the future.
We all know your game is buggy and some players would rather just use an older version. So when the update finally comes where the game is playable even with bugs (which has existed in the past pre-rewrite), players can enjoy it regardless of whether or not you support servers for that version.

Also you should add an option for the lobby host to kick players and re-add the console menu and potentionally a noclip mode for when players get stuck. (obviously behind a cheat/console mode like how the original game did it).