The Walkman was almost never called the Walkman outside Japan.
Sony's player went on sale in Japan on July 1, 1979, as the TPS-L2. It cost about 33,000 yen. Wikipedia describes the name as wasei-eigo, an English-sounding term made in Japan. It plays on the Pressman, a small Sony recorder sold to reporters.
The prototype came from the Pressman. Low End Mac says Sony co-founder Masaru Ibuka, who found his own recorder too heavy for travel, asked for a smaller player. The audio division removed the recording function and added stereo sound. Walkman Archive and Low End Mac both credit Kozo Ohsone, who led the division, with suggesting Walkman as a riff on Pressman. Wikipedia does not name him and instead names executive Norio Ohga as the one asked to design the playback-only player, so the credit rests on those two sources.
Not everyone liked it. The same accounts say company leadership doubted the name would catch on in the United States and Europe. Wikipedia says Sony's overseas sales companies objected to it as well.
So the player went out under local names. It was Soundabout in the United States, Stowaway in the United Kingdom, and Freestyle in Sweden and Australia. The Victoria and Albert Museum says Sony America had first proposed "Disco Jogger," which Chairman Akio Morita rejected because "disco" might put off older buyers. Soundabout, a play on "walk about," replaced it.
Meanwhile the Japanese name spread by itself. Visitors to Japan bought the player as a gift. One story says the name stuck after Morita visited Sony employees in Paris and their children asked him when they could get their Walkman.
By late 1980, Sony had given up the local names. The V&A says Morita declared Walkman the worldwide standard in April 1980, and Wikipedia says global branding began that November. The sources differ on the month.
Walkman also followed an older English pattern. Etymonline records walkie-talkie in 1939, built from the verbs walk and talk.