Do the "set ability score to 19" magic items function while using Wild Shape?

Question

If you, a druid who is attuned to an Amulet of Health, Gauntlets of Ogre Power, or Headband of Intellect, uses Wild Shape, choosing to wear the piece of equipment, do you gain the benefit of the increase?

For reference, here are the relevant descriptions:

[Name]

Wondrous item, [rarity] (requires attunement)

Your [ability] score is 19 while you wear this [item type]. It has no effect on you if your [ability] is 19 or higher without it.

Wild Shape

Starting at 2nd level, you can use your action to magically assume the shape of a beast that you have seen before...

While you are transformed, the following rules apply:

• Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores...
• You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so...
• You choose whether your equipment falls to the ground in your space, merges into your new form, or is worn by it. Worn equipment functions as normal, but the DM decides whether it is practical for the new form to wear a piece of equipment, based on the creature's shape and size. Your equipment doesn't change size or shape to match the new form, and any equipment that the new form can't wear must either fall to the ground or merge with it. Equipment that merges with the form has no effect until you leave the form.

Answer 1

Depending on the shape, yes

For the item to function, the wild shape must be able to wear it (emphasis mine):

Your [ability] score is 19 while you wear this [item type].

When the druid assumes their wild shape, they determine whether they wish to continue wearing the item. Any other choices (dropping it or having it merge with their new form) will not satisfy the "while you wear this" requirement. Further, the rules for wild shape explicitly say that merged items do not function (emphases mine).

You choose whether your equipment falls to the ground in your space, merges into your new form, or is worn by it. Worn equipment functions as normal, but the DM decides whether it is practical for the new form to wear a piece of equipment, based on the creature's shape and size. Your equipment doesn't change size or shape to match the new form, and any equipment that the new form can't wear must either fall to the ground or merge with it. Equipment that merges with the form has no effect until you leave the form.

Also as cited above, the DM determines whether the new form can continue to wear the equipment, "based on the creature's shape and size". We are also told that the equipment itself doesn't change size or shape, but it is important to realize that this is a general rule that assumes mundane equipment. In 5e, magic items meant to be worn are a special exception:

In most cases, a magic item that's meant to be worn can fit a creature regardless of size or build. Many magic garments are made to be easily adjustable, or they magically adjust themselves to the wearer.

So the size of the new form itself, whether a garden spider or a cave bear, is not an issue. The amulet, gauntlets, or headband will magically resize themselves, should the druid choose to continue to wear them.

However, the new form must have equivalent anatomy - in this case, a neck, hands, or a head (ibid):

A magic item meant to be worn must be donned in the intended fashion: boots go on the feet, gloves on the hands, hats and helmets on the head, and rings on the finger...When a nonhumanoid tries to wear an item, use your discretion as to whether the item functions as intended. A ring placed on a tentacle might work, but a creature with a snakelike tail instead of legs can't wear boots.

Most wild shapes adopted by a druid will have a neck or a head and thus could continue to use the amulet or headband. The more difficult one will be hands to wear the gauntlets. An ape has hands (including thumbs) and so could wear the gauntlets and retain their use, but most other wild shapes would not have appropriate anatomy and so would not be permitted to wear the item.