Green’s Dictionary of Slang

Beecham’s (Pills) n.

also pills
[despite Partridge, as echoed by Maledicta IV (1980), more likely a play on pill n. (1b)/ Beecham’s Pills, the popular UK medicine, than rhy. sl. (on testicles)]

1. the testicles.

[UK]J. Franklyn Dict. of Rhy. Sl. 37/2: Beecham’s Pills (2) testicles [...] invariably reduced to ‘pills’, demands for its rhyme the pronunciation, ‘testi-kils’. It is employed extensively at several social levels, generally in the setting ‘Don’t talk pills’ etc.
[UK]Dodson & Saczek Dict. of Cockney Rhy. Sl.
[US]Maledicta IV:2 (Winter) 196: Beecham’s (pills) as rhyming slang on testikills (!) and flowers (and frolics) or fun and frolics as rhyming slang on bollicks are rarer.
[UK]Partridge DSUE (8th edn) 64/1: Rhyming s. for ‘testicles,’ mispronounced testikills, late C.19–20.

2. in ext. use, nonsense [i.e. balls n. (4)].

see sense 1.

3. cartridges.

[UK]Yorks Eve. Post 16 Oct. 5/4: Most soldiers speak of their rifle, bayonet and cartridges as [...] ‘bond-hook’, ‘tooth pick’ and ‘Beecham’s pills’.