Is there an English word that adequately describes the action of bringing the subconscious memory to the conscious?
For example, showing a watch to someone that makes them remember a forgotten event from long ago.
Perhaps you are looking for "spark off", "trigger", or "jog one's memory".
According to MacMillan Dictionary
to make someone feel, think of, or remember something
Their growing reputation had sparked interest from other film-makers.
Your article sparked off happy memories for me.
This link points to some synonyms for cause someone to remember
The word isn't of English origin, of course, but the closest sensation I can compare to what you're describing is that of "deja vu." When one experiences deja vu, they have a very strong sensation of having experienced/been/done/seen this exact moment in time already. The expression is used commonly in the English language.
verb
[with object]
1.1 Cause one to remember or think of.
See also: Keim, "Forgotten Memories Are Still in Your Brain," wired.com
remind (ODO)
verb
[with object]
1 Cause (someone) to remember someone or something.
[with object and direct speech] ‘You had an accident,’ he reminded her