1) Navigate to ..\EA GAMES\Battlefield 1942\Mods\bf1942\Settings\ and open the Serversettings.con with Notepad.
2) Search for serverNameTagDistance and change the value after that to 0
3) Now the nametag only shows if you aim directly at the bot. But you can also delete the nametags completely, just change the value after serverNameTagDistanceScope to 0 as well.
That also works for Bf2/FH2 btw.