There were car phones in the fifties and maybe before. See them in old movies sometimes.
I would guess the service was very localised though.
I’ve seen pictures of them from the mid 1940’s on. The first phones, you had to call the radio operator first and give your callsign and IIRC, you had to say “over” to hear the other side talk. Later on, you had direct dialing. I used to pick them up in the 35/43 Mc, 152 Mc and 454 Mc bands on my police scanner. A lot of pagers were there too, especially 35 Mc, I used to hear the “bleep bleeps” and then the voice message. There is an amateur radio operator that even has recordings of pagers and mobile phone from the late 1960’s and 1970’s.