Yes, you can use the bonus action from Cunning Action in the middle of an Attack action
As you quoted in the question itself, the rules on bonus actions state (emphasis mine):
You choose when to take a bonus action during your turn, unless the bonus action's timing is specified, and anything that deprives you of your ability to take actions also prevents you from taking a bonus action.
In essence, you can take a bonus action at any time on your turn if its timing is not specified. Unless your ability to take the bonus action is dependent on a particular action being completed, nothing prevents it from happening in the middle of the multiple attacks of your Attack action.
The rogue's Cunning Action feature states:
[...] You can take a bonus action on each of your turns in combat. This action can be used only to take the Dash, Disengage, or Hide action.
The timing of the bonus action granted by the Cunning Action feature is not specified; it just unconditionally says that you can use a bonus action to Dash, Disengage, or Hide. As such, you can use the bonus action granted by Cunning Action at any time, including in the middle of an Attack action.
Rules designer Jeremy Crawford reiterated this fact, specifically referencing Cunning Action, in an unofficial tweet from May 2018:
[...] A bonus action that has no trigger—such as Cunning Action and the misty step spell—can take place whenever you want on your turn (PH, 189).
The clarification repeats what the rules indicate: bonus actions whose timing is not contingent on a specified action can be taken at any time, including in the middle of an action. Since Cunning Action's timing is not specified at all, you can take a bonus action granted by Cunning Action at any time on your turn.
(Earlier in the tweet, he clarifies that a previous tweet is meant to indicate that bonus actions and reactions that are dependent on the Attack action - unlike Cunning Action - can not interrupt the action, unless their trigger is an individual attack (or hit, etc.) itself rather than the Attack action as a whole.)