The feat specifically states it at the end of the description in the PHB (emphasis mine):
you do not gain the ability to speak while in a wild shape.
Furthermore, the feat clearly states that the verbal and somatic components are replaced by sounds and movements that the animal can normally make, such as the stated example of a hawk in the PHB:
For example, while in the form of a hawk, you could substitute
screeches and gestures with your talons for the normal verbal and
somatic components of a spell.
However, the question regarding the material components is a very interesting one as that is not explicitly stated in the feat text. The PHB text is as follows:
You can also use any material components or focuses you possess, even
if such items are melded within your current form.
To interpret the word use here we have to look at page 174 of the PHB which states what use means for both a material component and a focus (emphasis mine):
Material (M): A material component is one or more physical substances
or objects that are annihilated by the spell energies in the casting
process. Unless a cost is given for a material component, the cost is
negligible. Don’t bother to keep track of material components with
negligible cost. Assume you have all you need as long as you have your
spell component pouch.
The description clearly states that any material components are destroyed. Now let's look at the description regarding a focus (emphasis mine):
Focus (F): A focus component is a prop of some sort. Unlike a material
component, a focus is not consumed when the spell is cast and can be
reused. As with material components, the cost for a focus is
negligible unless a price is given. Assume that focus components of
negligible cost are in your spell component pouch.
Here the description clearly states the when using a focus, it is not consumed. Therefore, combining the above it is clear that the feat allows the caster to consume the material components that the original form had on him, as well as allowing the (re)-usage of a focus. This also maintains balance as giving them "the abilities required to fulfil the spell's requirements" would lead to balance issues with spells that use an expensive component (to prevent over use).
Summary
- Natural spell does not allow one to speak.
- Natural spell allows one to substitute verbal and somatic components with animal sounds & movements.
- Natural spell allows one to consume components that are carried in the original form.
- Natural spells allows one to (re)-use a spell focus.
Animal Handling skill probably won't help
The Animal Handling skill means you are proficient in controlling and nurturing domesticated animals. There are difference between tameable, captive-bred and domesticated animals. Even when animals are domesticated, you still need skill to work with them, but that skill doesn't give you supernatural powers neither allows you to control and command any wild and/or untrained animal you can encounter.
Animal Handling proficiency can't be a substitute for any magic spell (like Dominate Beast) or class feature (like Ranger's animal companion).
If you are a player, then ask your GM
It is the GM's job to decide, what check you should make. They might ask you "make the Wisdom (Animal Handling) check" in response to you announcing "I am trying to make this bear do X" (to run away, for instance). They also might ask you for a Nature check. Or Intimidate check. Or give you Advantage because you are proficient in Animal Handling. Or ask "How do you do that?" Or simply state "the bear didn't run away".
In the end of the day, it is completely up to the GM — despite the fact wild animals usually don't cooperate with people in real world, things might differ in the setting you are playing in. If you are the GM, and you choose the check can be made, the DC should depend on the announced action. "I try to scare this bear and make it flee" should be much easier than "I try to convince the bear to go with us and fight our enemies".
Normally, players don't "use" skills
In games like Pathfinder there are various tables, describing what you can and can not do with the respective skill. In D&D 5e, however, skill descriptions are loose by design, and the reason is it's the GM who decides what check a player should make.
In 5e players aren't supposed to announce "I use my Animal Handling skill", "I use my Survival skill" or "I use my Persuasion skill". Instead, a player should describe what their character does - "I try to calm down my horse", "I search for any tracks and footprints", "I say the elf guard we mean no harm" - and the GM might ask for a ability/skill check in order to resolve the situation:
The GM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure.
Sometimes, the GM might ask for an ability check using a specific skill — for example, “Make a Wisdom (Perception) check.”
For example, if a character attempts to climb up a dangerous cliff, the GM might ask for a Strength (Athletics) check.
See more information in Player's Handbook, page 174 - "Ability Checks".
Best Answer
Yes. The counterspell rules make no mention that the target must be in a recognisable form or anything like that. I can counterspell a human, I can counterspell a demon and I can counterspell a human who happens to be in the form of a lion, a bear or whatever else.