The Hut remains.
The common phrasing
the spell fails
refers to a spell's casting. For example, Counterspell reads:
You attempt to interrupt a creature in the process of casting a spell. If the creature is casting a spell of 3rd level or lower, its spell fails and has no effect. If it is casting a spell of 4th level or higher, make an ability check using your spellcasting ability. The DC equals 10 + the spell's level. On a success, the creature's spell fails and has no effect.
(emphasis mine), Whereas Dispel Magic reads:
Choose one creature, object, or magical effect within range. Any spell of 3rd level or lower on the target ends. For each spell of 4th level or higher on the target, make an ability check using your spellcasting ability. The DC equals 10 + the spell's level. On a successful check, the spell ends.
(emphasis mine). Once a spell is successfully cast in 5e, it's effect doesn't fail; it ends.
This wording is consistent throughout the rules as well. Some spells state circumstances in which they fail (i.e their effects do not take place), and many somehow state circumstances in which their effects end, either through their Duration or explicitly, in the spell effect's description.
A simple example is Call Lightning and the concentration rules (emphasises are mine):
3rd level Conjuration
Casting time: 1 Action
Range: 120 feet
Components: V, S
Duration: Concentration, up to 10 minutes
A storm cloud appears in the shape of a cylinder that is 10 feet tall with a 60-foot radius, centered on a point you can see within range directly above you. The spell fails if you can't see a point in the air where the storm cloud could appear (for example, if you are in a room that can't accommodate the cloud).
Some spells require you to maintain concentration in order to keep their magic active. If you lose concentration, such a spell ends.
The druid is not squeezed by the tiny hut. Therefore, the Dire Wolfed Druid might poke out of the sphere if there is not enough space for it, making him vulnerable to spells and attacks (albeit with some cover).