Police computer messaging emerged in the 1960s -- Teletypes, basic computer communications, 2-way radio and telephones were widely used.
2-Way radio and telephones were the most reliable and fastest method. Fast being relative.
Even to this day, the telephone is used to query certain records directly from DMV / DOJ
Back then, intercomms/telephone hotlines allowed officers and dispatcher to communicate to those that maintained and queried files.
Prior to "computers" - the telephone and police call boxes were used to communicate with the station.
Record checks and fingerprint checks were not instant, but obtainable.
DMV started collecting "fees" in 1936 - up til the 1960s DMV issued paperwork registration -- and well, times were different. Not that crimes didn't occur, but people were a bit more honest back then. Following the rules and doing what was right.
Here is a decent timeline of police tech
Forensic Science Timeline