So say I'm a demolitionist and I want to set up an explosive charge, knowing that my opponent will arrive later. I'll make the mechanics check to set up the bomb in a dark nook in a hallway, hide behind a crate, and then twenty minutes later when the Big Bad walks down the hallway I press the button on the detonator. How does this work? When do I make the check, when I'm making the bomb or when I press the button? What sort of mechanics check is it? If the Big Bad is a Nemesis, do I apply ranks of Adversary to the mechanics check? Or is this a tricky way to get around it?

I'd say button push, and Adversary would definitely be added.

It really depends on the circumstances, but in your instance here I would probably have you roll the narrative elements into the one check.

you first need to make sure that your target is within the blast radius, so whatever means you use to determine or direct his path (pretty much any social or knowledge skill might get you there, as well as particularly reliable intel in the narrative) would be enacted before you make the Mechanics check with the explosives.

Then, the setting up and blowing up of the explosives would be all rolled into one Mechanics check. You make the skill check, and then once it's rolled you narrate what your character does from setting the charges, the detonator, hiding nearby, and then setting them off.

That's probably how I'd run it, especially because the dice really lend themselves to that kind of front-loaded rolling, giving you fuel for the narrative with their results. But it's just one of dozens of ways you could do it.

Few different things to also consider:

Are your charges intended to kill from the blast, propel shrapnel, or bring the house down? Are they designed for this application? Are they prepped and installed in a way favorable to this?

Second, what's your trigger? Tripwire? Remote? Proximity sensor? All have their strengths and weaknesses.

Finally, based on how you answered the above questions, what elements do you have control over (Mechanics), which can you accurately predict based on your experience and knowledge (Knowledge (Engineering)), and which do you have to trust to the manufacturers of the items and the nature of the beast (covered by item stats).

Anything more complicated than a simple bomb with a remote detonator or a timer will need a variety of checks and it should be at least a 2 or 3 step process: evaluate the situation to determine best placement (optional, could grant bonus or penalty to further steps), actually set the trap (using results to determine victim difficulties), and once that's done, it essentially becomes a triggerable hazard for the target, no more input needed from you. It's a hazard that they should have a chance to detect (proximity sensors can be detected with energy sensors, tripwires with a good eye, etc.), and if they're missed by the target and they fulfill the trigger mechanisms triggering condition, they fall victim to the trap. If it's direct (bomb explodes in your face) apply the bomb's damage as an attack, with the PC's mechanics results acting as their attack. If it's designed to bring down the house, run the bomb as attacking an object (the building), and suddenly the hazard (falling building) is much harder for the target to dodge, and bits of house do a good bit of damage, not to mention possibly pinning down or burying survivors if there are any.

Finally, based on how you answered the above questions, what elements do you have control over (Mechanics), which can you accurately predict based on your experience and knowledge (Knowledge (Engineering)...

I would advise against creating new skills. Knowledge (Education) for structural engineering, or Knowledge (Warfare) for tactical placement, would be sound.

So say I'm a demolitionist and I want to set up an explosive charge, knowing that my opponent will arrive later. I'll make the mechanics check to set up the bomb in a dark nook in a hallway, hide behind a crate, and then twenty minutes later when the Big Bad walks down the hallway I press the button on the detonator. How does this work? When do I make the check, when I'm making the bomb or when I press the button? What sort of mechanics check is it? If the Big Bad is a Nemesis, do I apply ranks of Adversary to the mechanics check? Or is this a tricky way to get around it?

Per the explosives rules in DC...

You roll a mechanics check when assembling the bomb, difficulty is Easy +1 per additional charge. Success will add 1 damage per to the detonation when you push the button.

You plant the explosive (probably a stealth check, but skulduggery would probably work too).

The Target approaches, likely making a Perception check to spot the weapon.

You set off the bomb, and everything within the blast radius takes damage. No critting or anything like that, just damage.

Now... would the GM be outta line to make you roll a Cool check to push the button and apply the Adversary ranks to it as a way of determining the "attack"? Eh, probably not....

Preferably, they work once you've cleared the blast radius.

Finally, based on how you answered the above questions, what elements do you have control over (Mechanics), which can you accurately predict based on your experience and knowledge (Knowledge (Engineering)...

I would advise against creating new skills. Knowledge (Education) for structural engineering, or Knowledge (Warfare) for tactical placement, would be sound.

Quite right. WotC Knowledge skills are not easily displaced in favor of FFG Knowledge skills...

Planted explosives are not weapons and as such work quite differently when compared to mines and grenades. They also usually have the downside (or upside depending) that they do a lot of damage to a lot of things, intended or not. Which is also why you don't treat them as normal attacks.

You make the Easy Mechanics check and succeed or fail. I recommend that the check is made in secret. That determines the damage and if it goes off at all. A failure with a advantages might give the demolitionist an opportunity to realize the improper set up.

You won't have adversary modifying the mechanics check regardless; as although the BBEG is your intended, they don't affect your setup. The issues if adversary does come into play become glaringly obvious if you don't actually know the BBEG is going to be your target.

If you are trying to not be obvious about the explosives, then when the target/s enter the area you'd check the applicable skills in which case adversary would indeed apply.

stealth would be used to hide an item from obvious site.

Off the top of my head, I think I’d be inclined to do it as two checks, minimum.

One check (or more) to set the explosive. You could have some other checks also added in, if you can explain how those talents/skills come into play — like understanding of architecture and engineering so as to know the right place to put the explosive, etc…. Fail this, and it could blow up on you while you’re trying to place it.

The second check would be when the target comes into range, and here I definitely would include things like Adversary from a BBEG, or whatever. And if you’re manually detonating with a remote, then you definitely need to roll your Cool in order to press the button at the right time. Fail this, and the explosive goes off too soon or too late, or maybe doesn’t go off at all.

But that’s just my seat-of-the-pants answer. To give anything more definitive, I’d have to re-read the rules to see what they have to say.