James Kraus
Automatic Electric Mobiltel car phone brochure, 1960
In the early decades of motoring; if you wanted to make a phone call while on the road, you pulled over, parked and fed coins into a public telephone.
Civilian in-car mobile telephones were first introduced in the 1940s. By the 1960s new models were developed incorporating significant improvements. Chief among these was the general rollout of full duplex operation allowing both parties to speak simultaneously. Full duplex phones obviated the need for a push-to-talk button in the handset and over and out-style communication. Another major advance, direct dialling, became available by mid-decade in selected markets.
1960 Desoto Fireflite with General Elecric direct-dial, full duplex mobile phone
Early mobile phones were radio-telephones, operating over radio frequencies. They were quite similar to police radios, but with conventional handsets as opposed to a microphone and loudspeaker. This allowed at least a modicum of privacy when other passengers were in the vehicle.
James Bond answers a call from headquarters via a phone mounted in his 3.5 litre Bentley in From Russia With Love, 1963
The phone unit seen in the interior was merely the handset and control head; the main transceiver units were so large that they had to be mounted in the luggage compartment.
1964 Chevrolet Impala Convertible with Automatic Electric Starlight phone
Mobile phones were extremely expensive, often costing as much as a medium-price car. Nonetheless, waiting lists of over a year were not uncommon.
Direct dial phone in Mercedes-Benz 230 S
In the early 1960s, mobile telephones were still largely vacuum tube or hybrid designs but by the end of the decade they increasingly began incorporating more transistor technology, greatly reducing the size and weight of the transceiver chassis as well as increasing reliability and lowering power consumption.
1965 Ford Galaxie with Secode direct-dial unit
As an option, radio-telephones could be connected to a relay that would sound the car horn when an incoming call was received in an unoccupied vehicle.
1966 Chevrolet Impala with ITT-Secode phone
Phones were a common addition to Rolls-Royces, Mercedes-Benz 600s and Cadillac Fleetwood 75s. Beginning with the 1963 Motorola MTS, car phones for limousines with partition windows became available, with rear-seat extensions allowing the chauffeur to originate or screen calls before forwarding them to the rear compartment.
The limited-edition 1967 Ford Thunderbird Apollo featured an Automatic Electric Starlight phone mounted just above a Philco rear seat television, providing easy access to all occupants.
Car phones were a staple of 1960s TV detectives, secret agents and superheros; used by Peter Gunn, Batman, The Green Hornet, The Baron, Mannix, Pete Cochran of The Mod Squad, and Steve McGarrett in Hawaii Five-O.
Los Angeles private eye Joe Mannix talks into a Motorola MJ Series phone in his 1968 Dodge Dart GTS
After the 1960s, on-hook dialling and speakerphone capability were introduced, and that was about it as far as development would progress in terms of the classic automotive radio-telephone as manufacturers switched their attentions to the burgeoning new cellular technology in the 1980s.
Auto Universum 
I was fortunate enough to use analog car phones on occasion in the 70s. I believe the maximum capacity of the mobile phone network in NYC was only 36 simultaneous calls.
A famous French film, Le Corniaud contains a few scenes involving car phones. In particular the scene where the hero, played by French actor André Bourvil, calls his ‘benefactor’ played by Louis de Funes from a Cadillac at a petrol station, thinking he is speaking to someone in Paris whilst his contact is actually hidden under a blanket in the Jaguar next to him! More about the film (and car phones in the movie) here: https://crankhandleblog.com/articles/le-corniaud-gangster-un-capolavoro-francese/)