Nav files don't get downloaded to the client 'cause they are for bots, which are run by the server. If a nav file is being sent to the clients, the server operator is doing something to cause that.
Don't use Pakrat it breaks your BSP so you can't ever pack anything in to it again. use VIDE.
No, nav files as well as bsp downloads to client if it's available. If server uses sv_downloadurl for fast downloads, then if the nav isn't uploaded to that host but is on the server, it won't download either (unless it is uploaded to the host too). But if the files come from the server directly, the nav will download to clients. Every time.
Nav file can be packed within the bsp and workshop upload even requires it but
thing with workshop is that nav files do not load at all from workshop bsp file if map is started from the game menu or with command map workshop/id. Valve needs to fix this because it's not just bots that fail, it's also the halloween bosses that people add to their maps due to nav failing to work.
But if you take the map bsp off from it's workshop dir and load it up in tf/maps folder, it (the nav) works fine.