Verb meaning "to alter someone's famous saying"

You say you are looking for 'way of saying that you are slightly, but intentionally, modifying a famous phrase'. Breaking that down, it seems that the modification would be obvious, leaving the need to clarify that it has been made deliberately and not through ignorance.

If this is correct, I would suggest that 'Evoke' would work in your sample sentence

Bring or recall (a feeling, memory, or image) to the conscious mind. ‘the sight evoked pleasant memories of his childhood’

hence

"To evoke Churchill, we are about to see 'blood, sweat and spears'"

I think you may use paraphrase:

• to repeat something written or spoken using different words, often in a humorous form or in a simpler and shorter form that makes the original meaning clearer.

(Cambridge Dictionary)

• To paraphrase Churchill , we are about to to see "blood, sweat and spears".