You can use the tattoo
The rules for Wild Shape state:
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.
However the rules for attuning to a tattoo also state (emphasis added):
To attune to this item, you hold the needle to your skin where you want the tattoo to appear, pressing the needle there throughout the attunement process. When the attunement is complete, the needle turns into the ink that becomes the tattoo, which appears on the skin.
The substance of the tattoo is the ink just like the substance of a scarf is the fabric. Just as a scarf lay on the form of a wolf, the ink that composes the tattoo has some flexibility in conforming to the surface of your skin. This means for a wide variety of tattoo shapes and wild shape forms the tattoo can fit just fine. For example a circular shield tattoo for a Small Barrier Tattoo on your scalp could reasonably fit on the scalp of a wolf. Considering that:
When applying a magic tattoo, a creature can customize the tattoo's appearance. A magic tattoo can look like a brand, scarification, a birthmark, patterns of scale, or any other cosmetic alteration.
A druid with tattoos could reasonably choose designs that can fit onto other forms with minimal distortion.
Beyond this there is no evidence that forcibly changing the shape of the ink would render the tattoo inoperable. If fact the rules for magical tattoos specifically state:
Once inscribed on a creature's body, damage or injury doesn't impair the tattoo's function, even if the tattoo is defaced.
This indicates that if the pattern of the tattoo were to deform or even break due to Wild Shape the magic of the tattoo would continue to function.
Lastly there is precedence for magical equipment resizing itself to fit the form of its user (DMG, p. 140):
In most cases, a magic item that's meant to be worn can fit a creature regardless of its size or build. Many magic garments are made to be easily adjustable, or they magically adjust themselves to the wearer.
As discussed in this answer there needs to be a ruling for the particular magical item as to whether it does indeed resize itself and whether it does so automatically or whether it is merely easy to adjust. In the case of magical tattoos we already know that the tattoo can fit any creature at the time of attunement so the question is whether it would continue to match the creature during changes in its body shape.