Do I and the target have to wear the platinum rings at casting time of Warding Bond?

The material components of Warding Bond are (emphasis mine)

*- (a pair of platinum rings worth at least 50 gp each, which you and the target must wear for the duration)

Since the description clearly states that the two creatures must wear such rings for the duration, and the duration begins whenever the spell's effects begin, is not clear to me if during the casting the cleric$$\^\star\$$ and the target must already wear the rings.

The only aspect that I am sure of is that the cleric$$\^\star\$$ must have (in some way, on its finger or holding in the hand) the ring at casting time, per material components rules. It is true that during the casting they may put on the ring, hence satisfying the requirement. For what regards the target, must they wear the ring at casting time?

This could be not a problem outside combat, but during a fight the action economy may draw some attention to this aspect of the spell.

Suppose that Eddy the Artificer is fighting alongside Adrian the Fighter, who is severely injured but has to pass through several enemies in the battlefield (receiving hence several opportunity attacks) for reaching with the Dash action the lever that opens the drawbridge and make enter the reinforcements. Eddy starts to cast Warding Bond, but Adrian is not wearing the ring Eddy gave to him, because it is too fancy&showy. Assuming that Adrian can easily take the ring from its pocket, how they can resolve the situation for succesfully cast Warding Bond? Do they both need to wear the rings at casting time?

$$\^\star\$$ Or the paladin or the artificer or any caster that gains access to this spell.

You do not need the ring at casting time

Your observation is right. There is no mention of needing to wear the ring at casting time, only during its duration, so in theory you could cast the spell without wearing the ring. It would only fail if you do not wear the ring during its duration.

However, there's one little issue that arises from doing this...

The caster cannot cast and put on the ring at the same time

The casting time of Warding Bond is one action. Even if putting on the ring is done as a free item interaction, actions are atomic in 5th edition, meaning you cannot break them down, or do half an action, do something else then finish your action.

Because of that, you either have it already equiped before casting, or you don't. In that second situation, the spell's duration starts while you do not have the ring equiped, and the spell ends instantly.

Of course, a DM could rule otherwise and let you equip your ring while casting. The somatic components of the spell are not explicitely stated, so it's up to the DM to decide what they are and if they can include putting on the ring, as well as if you'll have to use your free item interaction in order to do so.

Using the Ready action solves the issue... at a cost

If you ready your spell instead of casting it normally, its duration does not start until you release it. You can then put on the ring and release the spell, starting its duration and having it behave as intended.

If your target is not wearing its ring, instead of releasing the spell when you don your own ring, you can have the trigger of the readied spell be something like "when the target dons their ring", something they can most likely do with their free item interaction.

Do note that this uses both your action, your free item interaction and your reaction, which seems like quite expensive for your action economy.