Green’s Dictionary of Slang

blower n.2

1. [early 19C] a pipe.

2. [mid-19C] (US) a liar.

3. [mid-19C–1920s] a braggart; also attrib. [blow v.1 (1e)].

4. [1920s+] a telephone; esp. in phr. on the blower [following on f. the earlier ‘speaking tubes’ down which one had to blow to alert the other person; note bookmaker jargon blower, the betting shop public address system that broadcasts races, odds and results].

5. [1950s] (US) a broken-down horse [NB Aus. bookmaker j. blower, a horse, initially a favourite, whose odds lengthen].

6. [1990s+] an annoying person.

In phrases

cop for the blower (v.) [cop v. (1d)]

[1960s] to make a phone call.