The transformation is magical, the shape itself is not
As NautArch first answered, the relevant source is the Circle of the Moon Druid's ability:
Starting at 6th level, your attacks in beast form count as magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage.
It is a basic assumption that if a specific feature explicitly grants an ability, other similar features do not grant it unless they explicitly say so as well. So if Primal Strike says its beast form attacks are magical, we can safely assume that they are not magical in other druid wild shapes.
In this case, the process is what is magical, although the result is not. The druid is using magic to transform into a beast, but the beast they become is not magical. It is a more open question as to whether there is an ongoing magical effect which sustains them in the non-magical form of the beast. I would say not, but I don't think there is strong evidence either way.
I have been asked by NautArch to elaborate on this:
Transformation vs. Shape
You correctly cite the Sage Advice Compendium as providing the litmus test for whether something is magical. It is clear that a Druid's Wild Shape is not a magic item, not a spell or recreation of a spell, not a spell attack, and is not fueled by the use of spell slots. The only question is, then,
- Does its description say it’s magical?
(where "it" is "a game feature")
So, does the description of the Druid's magical class ability Wild Shape say that the shape assumed is magical? It does not.
Starting at 2nd level, you can use your action to magically assume the shape of a beast that you have seen before.
Note that it does not use "magical", an adjective that could modify 'shape'. Rather, it uses "magically", an adverb that modifies the verb 'assume'. That is, the transformation, or assumption of the shape, is magical. But it does not say the shape itself is magical. It does not say
you can use your action to assume the magical shape of a beast
Further evidence of the non-magical nature of the shape is provided by the undisputed answer to your question; a wild-shaped druid's natural attacks do not count as magical for the purpose of magic resistance.
The Monster Manual Errata specifies that
"Particular creatures are even resistant or immune to damage from nonmagical attacks (a magical attack is an attack delivered by a spell, a magic item, or another magical source).
We know, thanks to the Moon Druid's Primal Strike, that most wild shapes do not count as a 'magical source' of damage, because if they did, their attacks would count as magical. To argue then that the druid's wild shape is magical, one would need to explain why the shape itself is magical, the shape is the source of the attacks, but yet attacks from the shape are not delivered by a magical source.
But does a magical effect sustain the druid in beast form?
This is more open to interpretation. Certainly the description of the ability does not indicate that anything more than the transformation itself is magical. It does not say
you can use your action to magically assume and sustain the shape of a beast
And it is possible for the process of transformation to be magical even when the result is not and no magic sustains the transformation. This is perhaps best described in the March 2016 Sage Advice (although talking about the results of a spell rather than a magical class feature, emphasis mine):
Can you use dispel magic on the creations of a spell like animate dead or affect those creations with antimagic field? Whenever you wonder whether a spell’s effects can be dispelled or suspended, you need to answer one question: is the spell’s duration instantaneous? If the answer is yes, there is nothing to dispel or suspend. Here’s why: the effects of an instantaneous spell are brought into being by magic, but the effects aren’t sustained by magic (see PH, 203). The magic flares for a split second and then vanishes. For example, the instantaneous spell animate dead harnesses magical energy to turn a corpse or a pile of bones into an undead creature. That necromantic magic is present for an instant and is then gone. The resulting undead now exists without the magic’s help. Casting dispel magic on the creature can’t end its mockery of life, and the undead can wander into an antimagic field with no adverse effect.
Similarly, it could be that the magic of the druid's transformation flares for instant, but is then gone, leaving the druid in the non-magical shape of a beast. Although the shape has a limited duration, the transformation itself is instantaneous and is thus not sustained by magic.
Or, it could be that although the beast shape is not itself magical, the druid is held in that form by a magical effect.
This could be 'tested' by the druid entering an antimagic field: A druid in an antimagic field could not transform into their wild shape (or presumably back) in any event. If a druid already in wild shape could walk into an antimagic field without having their shape removed, then there is no magical effect sustaining the transformation. If being in an antimagic field causes a already transformed druid to lose their wild shape, then the shape itself is sustained by magic (See also: "Does the shapechanger trait work in an antimagic field?).
This could also be tested by casting detect magic on the wild-shaped druid. The spell allows you to "sense the presence of magic", and thus if the druid's shape was maintained by an ongoing magical effect, it could be detected (See also: "Does Detect Magic detect supernatural creatures like demons or items connected with them?" and "In what ways can a druid's Wild Shape be detected?")
Unfortunately, I don't know of any official rulings on these potential tests. To answer what would happen when applying these tests (see the links above), one has to make an assumption about whether an ongoing effect is present, rather than having an official source say that an ongoing effect is present, and the results of the tests being derived from that.
If we admit unofficial rulings, Jeremy Crawford clearly believes that Wild Shape is an ongoing magical effect. In this tweet (thanks to Naut Arch) from May of 2016, Crawford says that Wild Shape would be suspended by an antimagic field. And in this tweet from a week later he says that a druid in Wild Shape would be detected by detect magic, thus hitting both of our test cases. For context, the test for whether or not a game feature is magical was published in Sage Advice Version 1.08 in February of 2016, three months previous, so Crawford was presumably operating under its principles when he wrote those tweets.
Absent official evidence to the contrary, I personally think that there is not a magical effect sustaining the change, simply because the description of the feature itself does not mention one. Parsimony suggests that if the transformation, but nothing else, is cited as magical, then that is the only thing that is magical.
Others, such as NautArch (comments herein) and Lino Frank Ciaralli (comments here), disagree, perhaps based on their personal interpretation of 'how magic works'. There is certainly unofficial support for this position from two clear Crawford tweets.