The rules on Two-Weapon Fighting state:

When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand.

However, the description of bonus actions states:

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

So, can you attack with your bonus action first?

You can't take the bonus-action attack before the Attack action, because the bonus-action attack is conditional upon you taking that action.

You can't take the bonus-action attack before the Attack action (in which you attack with a one-handed light weapon), because the bonus-action attack is conditional upon you taking that action.

Jeremy Crawford addresses this on Twitter here (prompted by the wording of the Shield Master feat):

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.

This contradicts a previous ruling by him via Twitter, in which he stated that, "As with most bonus actions, you choose the timing, so the Shield Master shove can come before or after the Attack action." Crawford acknowledges this change here:

In 2017, I changed the ruling on bonus action timing because the old ruling was illogical. The original ruling failed to account for the fact that X relying on Y is a form of timing. The new ruling corrects that oversight.

And further explains his reasoning here:

The old ruling on bonus action timing didn't quiet questions on that timing. Instead, the illogical ruling fueled questions, and it even inadvertently led some fans to think our choice of words like "if" or "when" had super-precise meanings in bonus actions. They don't.

As you quote, the Two-Weapon Fighting rule says:

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

The bonus-action attack is conditional upon you using the Attack action and attacking with a qualifying weapon. As such, you can't use your bonus-action attack before you've taken the action that qualifies you for it.

Note that 'Attack' doesn't require you to use the same hand each time. If you want to use the other weapon first, just use that hand for your Attack action and the other for your Bonus action. This works for your example of two simple martial weapons. In the Shield Master example, you'd still need to attack first.