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My Fighter (Battlemaster) Warlock (Hexblade) build character has just acquired a suit of +1 adamantine plate armour.

From the description of adamantine armor (DMG, p. 150):

This suit of armor is reinforced with adamantine, one of the hardest substances in existence. While you're wearing it, any critical hit against you becomes a normal hit.

Is a roll of a natural 20 (which is normally a critical hit) still automatically a hit, despite the critical becoming a "normal hit" because of the armour? Or would the attacker need to exceed my AC in order to score the "normal hit" if (for instance) I cast shield or use the Evasive Footwork maneuver to boost AC?

My AC is 20, so with shield cast it becomes 25; for example, would a goblin with a +4 attack modifier score a hit on a roll of 20 against me? A total of 24 would not be sufficient to "hit" under normal rules, but does the "20 is always a hit" mechanism override this despite the critical being cancelled by the adamantine armour?

Yes, the natural 20 is still an automatic hit

Jeremy Crawford, the 5e lead rules designer, made an official ruling over twitter saying as much:

Q: Does the nat 20 still auto hit against adamantine armor?

A: Yes.

The language of the item also supports this ruling. As you quoted, the armor states:

While you're wearing it, any critical hit against you becomes a normal hit.

A critical hit is, by definition, an automatic hit. The item doesn't say that the attack is a miss unless it exceeds your armor class even with a natural twenty. It says that critical hits against you become normal hits.

So, that suggests the following sequence of events:

1. Attack roll is made against you


2. The roll is a natural 20, which means it's an automatic hit


3. The effect of the armor kicks in, the critical hit becomes a regular hit.

By the time the item interferes, we've already established the attack as a hit and the armor doesn't undo that hit; it reduces the severity of the hit.

But What about for Champion Fighters?

Champion Fighters have class features which give them an expanded critical hit range

Beginning when you choose this archetype at 3rd level, your weapon attacks score a critical hit on a roll of 19 or 20.

Starting at 15th level, your weapon attacks score a critical hit on a roll of 18–20.

It's already been established in a related question that these improved criticals are also automatic hits

Improved Critical specifically says you score a critical hit on a 19 or 20. A critical hit is a type of hit; by scoring one, you've also scored a hit. If the word 'critical' wasn't there, the ability would certainly read as if you couldn't miss on a 19 or 20.

So, the attack is a critical hit, which means that the adamantine armor makes it a confirmed normal hit. So, if you face any creatures with improved critical hit ranges, they will still automatically hit you on a natural roll that falls within their range. However, those automatic hits become regular hits just like a critical hit from a natural 20.

Improved Criticals Sage Advice

Any critical is treated as a normal hit in regards to the damage dice.

Indeterminate, and it opens another can of worms...

There is some legitimate dispute on this one. The Basic Rules have the following passage:

Rolling 1 or 20. [...]If the d20 roll for an attack is a 20, the attack hits regardless of any modifiers or the target's AC. This is called a critical hit, which is explained later in this section.

Later on in the Damage Rolls section, it describes the damage effects of a critical hit. It all seems clear at this point - a natural 20 being automatic hit and a critical hit are the same thing.

However, when we look at the champion fighter's Improved Critical ability, we have:

Improved Critical. Beginning when you choose this archetype at 3rd level, your weapon attacks score a critical hit on a roll of 19 or 20.

"Rolling 1 or 20" indicates that the die roll automatically hits, and calls it a critical hit. Champion fighter also calls a 19 or 20 a critical hit. Does that mean a 19 is also an automatic hit just like a 20?

Options

As much as I am loathe to say it, this is a DM discretion call. While there are some designer-intent posts on Twitter, they have not made it into an officially published errata. Considering the number of times those seemingly official posts have made a call then reverted it, they should all be considered suspect.

There are two logically internally, consistent choices:

1. 
   If the DM wishes to consider an "automatic hit" and the dice-doubling "critical hit" to be separate things.
   
   
   • On a 20, adamantine stops the damage ("critical hit"), but the "automatic hit" carries through because a 20 always hits.
   
   
   • Champion fighters automatically hit on 20s, but not 19s (a 19 is a "critical hit" but not an "automatic hit".).
   
   


2. 
   If the DM wishes to consider "automatic hit" and the dice-doubling "critical hit" as synonymous:
   
   
   • Adamantine stops the dice-doubling from the "critical hit" and, because they're the same thing, also stops the "automatic hit" (compare the total attack roll to the AC).
   
   
   • Champion fighters count 19s (and eventually 18s) as "critical hits", which means they are also "automatic hits".
   
   

To put it differently... any two of them (the combat rules, adamantine armor, and the champion fighter), isolated from the third, doesn't leave much room for doubt. It is the interaction of all three passages that creates ambiguity.

I've specifically avoided sharing my opinion, and presented two possible interpretations. The published rules don't provide sufficient clarity or weight to make an iron-clad determination either way.