Yes.
First off, let's note that there's nothing in the Druid description that specifically precludes the WS1 → WS2 transformation you're contemplating.
Second, consider this line of "Wild Shape":
You retain the benefit of any features from your class... and can use them if the new form is physically capable of doing so.
Wild shaping is a druid feature, so you retain it if the form is capable of doing so. Is it? For me there are 2(.5) reasons to suppose it is:
- Wild shaping does not require any spell-like components: gestures, utterances, &c. Some of those might have been impossible for your dire wolf, but they're not needed.
- We know a druid in animal form can transform: into a druid! The druid's personality, mental scores, and proficiencies all carried over. Absent a prohibition, it seems strange to think that this one part of a druid's core doesn't carry over.
- (Plus, if memory serves, Robyn--the druid queen of Douglas Niles Moonshae trilogies which informed much of the early D&D canon on druids--once changed from one form to another. The time she ran into Faerun's first peryton, IIRC. Can't put my hands on that book right now, though.)
Lastly, what would it harm to say "yes" to your player? The action-economy seems to be the only place where this could be exploitative. If we were to rule "no," then a druid has to use both their bonus action and their action to transform WS1 → druid → WS2. If we rule "yes" then a druid has to use their action to transform WS1 → WS2, leaving them "ahead" by a bonus action. In my opinion, that's a pretty small risk to run: the average druid doesn't have a lot they can do with an isolated bonus action.
Yes, if you have fangs.
You've already rightly pointed out that you have "appropriate fangs", but I wanted to expound of this a little bit to make sure future askers of this similar question understand that caveat. You need fangs to make the bonus action unarmed strike.
While shifted, you can use your elongated fangs to make an unarmed strike as a bonus action. If you hit with your fangs, you can deal piercing damage equal to 1d6 + your Strength modifier, instead of the bludgeoning damage normal for an unarmed strike. (Wayfinder's Guide to Eberron, p66)
Many longtooth shifters have canine traits that become more pronounced as they shift, but they might instead draw on tigers, hyenas, or other predators.
And then
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 (PHB p67)
So, yes, you CAN make an unarmed strike in wild-shape, as pointed out in this answer. But here's the catch: you don't have elongated fangs unless you assume a form that has elongated fangs. Wildshape limits your racial features if your new physical form is not capable of doing so. Furthermore, you adopt a new physical form, so you wouldn't have your previously elongated fangs, you would have new teeth. If you transform into a horse, you don't have elongated fangs.
So what this means is that you can't use your bonus action to make an unarmed strike because you don't have the prerequisite (using your elongated fangs). And even if you could, you wouldn't get the elongated fang damage because you also didn't hit "With your fangs".
But lets say you DO have fangs, like the aforementioned tiger. Then you absolutely could make that unarmed strike with fang damage, provided you were Shifted before you made the attack.
What are "Elongated Fangs"?
The "Elongated" fangs are relative to your humanoid form.
Many longtooth shifters have canine traits that become more pronounced as they shift.
So you have sharp canines as a humanoid, but they can't really bite things that well. When you shift, they are elongated. So as long as your new form has fangs, like most predators do, then you can use them. Don't think of elongated fangs as these huge dagger-like, snake-like fangs, but rather just elogated in relation to your humanoid canines (which are already pronounced). A tiger, for example, would have fangs that are elongated in relation to your humanoid fangs. Your DM will decide if your fangs are good enough to use, since the rules don't tell us which animals have fangs nor how long they need to be. We have to use our best judgement.
Best Answer
There is no conflict here. As you noted a wild shaped Druid keeps features from their class, race, or other source they have and can use them when they are wild shaped.
The correct order, depending on the leniency of the DM, would be to wild shape FIRST and then shift to get the bestial appearance and its bonuses so you're looking at 2 bonus actions in the order.
But nothing is stopping it from working.
Some issues might arise around what type of shifter you were and what you wild shaped into. While the Beasthide, Swiftstride and Wildhunt subtypes are OK, the Longtooth type might be restricted to certain wild shape forms to take advantage of its elongated fangs with their unarmed strike attack, again that's probably DM dependent.