Anyone wielding two light weapons can attack with one weapon and use a bonus action to make an offhand attack.

The Rogue's having Cunning Action as an option does not prevent them from instead choosing other bonus actions. The restriction only applies to Cunning Action itself - if the Rogue chooses to use Cunning Action, only Dash, Disengage, and Hide can be used. Otherwise they may use any other Bonus Action which is available to them. This distinction is there because the Thief and Arcane Trickster archetypes later expand the options provided by Cunning Action.

The bottom line:

1. The Rogue needs the Dual Wielder feat to use Two Weapon Fighting with a rapier, since the rapier is not a light weapon.
2. The Rogue can use either Cunning Action, or an offhand attack, but not both.
3. Having the Cunning Action class feature does not prevent the Rogue from choosing other bonus actions instead.

The language is unclear

The intent may have been that you must wait for your Action to complete before utilizing your Bonus Action.

Jeremy Crawford had previously tweeted that Bonus Action timing was completely up to the player regardless of the trigger requirement. While this gave a lot of freedom of choice, it wasn't really fully in tune with the written rules on Bonus Action timing (emphasis mine).

You choose when to take a bonus action during your turn, unless the bonus action's timing is specified...

However, that tweet has been updated after realizing his mistake.

Clarification about bonus actions: if a feature says you can do X as a bonus action if you do Y, you must do Y before you can do X. For Shield Master, that means the bonus action must come after the Attack action. You decide when it happens afterward that turn.

He further clarified with this tweet

No general rule allows you to insert a bonus action between attacks in a single action. You can interrupt a multiple-attack action with a bonus action/reaction only if the trigger of the bonus action/reaction is an attack, rather than the action.

But that clarification was...wait for it...further clarified again:

"My tweet below was addressing bonus actions and reactions that have triggers. 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 RAW and the Crawford

It seems clear that Crawford's intent for Bonus Action triggers is that there is a requirement for the triggering Action to fully complete before you can take the Bonus Action. If you go by this, then you are not able to take your Bonus Action inside of (interrupting) your Attack Action.

A question of wording

However, as stated above, GWM is not triggered by the completion of an Action, but by the killing of or reducing an enemy to 0 HP. Given that the trigger is not based on the Attack Action but on the result of a hit combined with the rule on choosing when to take a bonus action during your turn, then it seems possible to interrupt your Action.

Which way to go?

I understand what Crawford is saying in that Actions are complete events that can not be interrupted...except by movement. His clarification that you can't interrupt one action (Attack) with another (Bonus Action) makes some sense, but given the precedent of allowing movement inside of an Action and that the trigger for GWM is not the Action but an event, a GM could rule at their table to allow it.

Crawford's intent here is understandable, but it still doesn't make 100% logical sense given that you can already interrupt your action with movement and/or reactions.

At my table, I generally let my players shift and shuffle actions as long as they aren't actually limited by a clear mechanic. In this case, it makes more than good sense to allow the player this and I haven't seen any issues with it at my tables.