I was originally looking up the word 慎重 and came across the other words. All three of them are translated as "discretion." What's the subtle difference and/or nuance among all three of them?
用心 is mainly used as a suru-verb today, except in this idiomatic catchphrase and 用心深い. It refers to alertness against bad events, especially crimes.
慎重 (na-adj) and 用心深い (i-adj) are indeed similar in meaning, and sometimes interchangeable. The former is simpler and used whenever you need carefulness/alertness (like doing a math calculation or turning over a pancake). The latter is usually more about security and used with risks such as fire, thievery, fraud and natural disasters. 慎重な人 is a careful person who takes time before trying something new, whereas 用心深い人 is a guarded/cautious person who thinks they are at risk.
用心深さ is the level/degree of 用心, just as 慎重さ is the level/degree of 慎重. (See how to use さ)