Though Steam user tsupertsundere's guide is great for getting every achievement, if you're running Any%, their guide is a bit drawn out and covers unnecessary moves. Because of that, I wrote one that covers only the required actions for getting through the story as quickly as possible. Note that the sections here are not official names, but merely ones I came up with myself.

Time starts when you click "New Game".

Sneaking Out (Aidan)

1. Grab the shirt
2. Grab the stick while Dougal's mouth is open
3. Use the stick to grab the trousers
4. Give the sandwich to Dougal
5. Use the stick to make Dougal roll over
6. Throw the remainder of the sandwich at the sweater
7. Trade the stick for the sweater while Dougal's mouth is open
8. Get the boots
9. Sneak out via the ladder

Get Ready for the Day (Aidan)

1. Pick up the jug from behind Aidan
2. Go to the larger room view and pick up the footstool
3. Go to the back garden
4. Place the jug beneath the pump
5. Try to use the pump, breaking the handle
6. Use the broken handle to pry open the shed door, then enter the shed
7. Pick up the pitchfork on the ground
8. Return to the garden
9. Stab the pitchfork into the stump beside the pump
10. Use the makeshift hammer to run the pump
11. Pick up the jug and take it back inside
12. Go outside through the front
13. Look inside the letterbox
14. Grab the mail one by one until the package is opened
15. Read the enclosed letter
16. Take out the crystal
17. Go back inside

Wrong Way to Town (Aidan)

1. Snoop around in Arthur's room beside the ladder
2. Grab the mask from atop the bed
3. Investigate the machine
4. Pour the water into the funnel
5. Place the crystal into the slot
6. Go back to full dash view
7. Set the resistance (Ω) to 4 and the weight (lbs) to 3
8. Turn on lights (1,2), (2,3), (3,3), (3,4), (4,3) (origin is bottom-left). Visually, that is: ◻◻◼◻ ◻◼◼◼ ◼◻◻◻ ◻◻◻◻
9. Leave the room, picking up the mask if not done already
10. Return to the garden, then go to the shed
11. Turn on the lights with the switch near the entrance
12. Use the stool to retrieve the box from above the shelf on the far-right
13. Pick up the puzzle box
14. Leave the shed, then go back inside the house
15. Use the mask to scare Dougal
16. Use the puzzle box to start a conversation with Lily
17. Select, in order: both arms up, crouch, one arm up, upside-down (going clockwise: 3, 4, 2, 1)
18. Take out the key
19. Return to the shed
20. Use the key in the keyhole within the machine and be whisked away to Clonfira

Not on an Empty Stomach (Lily)

1. Check the table behind Lily
2. Grab the (empty) bag of oats
3. Pick up the pot behind Dougal
4. Point Dougal to the door, then quickly get the oats above the bookshelf
5. Get Dougal to remove the string keeping the cupboards closed
6. Open the cupboard and pick up the bowl
7. Check the table again and make breakfast

A Whole New World (Aidan)

1. Look at Arthur's belongings on the ledge
2. Take the hat and glass jar with you
3. Place the jar into the left socket, then quickly take it out and place it in the right socket
4. Pass through the gate and venture out into the unknown

A Short Work Day (Lily)

1. Go to the side of the house, where the tractor is
2. Try to remove the spanner in the hood of the tractor
3. After the hood flies open, look inside and grab the fan belt
4. Go to the coal bunker beside the house and take out the bone from inside it
5. Throw the "rope" over the tree branch on the right side of the map
6. Sit in the makeshift slingshot
7. Use the bone to get Dougal to pull you back
8. After the shop scene, attack the gnome with the wooden sword
9. With the gnome out of the way, reenter the house

Uncharted Territory (Aidan)

1. Interact with the lamp
2. Cross the bridge that appeared
3. Touch the first lamp you pass by
4. Ignore the lamp by the door and touch the blue lamp instead
5. Now that the door is open, go through and explore the unknown

Pictorial guide to Steps 3–5: https://imgur.com/fqw9Rbv

Where's Dad? (Lily)

1. Snoop around in Arthur's room
2. Collect Sir Fuzzle Puff
3. Leave the room and go out to the shed
4. Send Sir Fuzzle Puff ahead in the Adventure Machine
5. Pick up a piece of paper beside Dougal
6. Attach a Note to your poor message-board doggo Dougal
7. Enter the Adventure Machine and go on an adventure of your own

We Need to Go Deeper (Aidan)

1. Interact with the lamp
2. Touch the lamp that's lowest on the screen
3. Go to the lamp on the left side of the screen
4. Go back and touch the lamp directly across from the entrance
5. Go by the door and touch that lamp
6. Use the blue lamp to open the door
7. Cross the middle platform to the leftmost lamp, and touch it again
8. Touch again the lamp directly across from the entrance
9. Pass through the door

Pictorial guide to Steps 2–9: https://imgur.com/JhQEyOY

An Adventure of Her Own (Lily)

1. Leave the entrance to Clonfira, following Aidan into adventure
2. As Aidan, cross the bridge and meet Merr
3. As Lily, touch the orb, destroying it instead
4. Cross the bridge and the island it connects to, only to be picked up by a monster
5. Use the Sword to give the monster's tongue a good whackin'

Betrayed! (Aidan)

1. After the dialogue with Merr, follow him into the shrine
2. Pick up the glove lying in the corner of the room
3. In the side room, equip the glove and shoot the blue light
4. Equip the powered-up glove and touch the machine in the main room
5. Pick up the screwdriver near where the glove was
6. Return to the side room and use the screwdriver to plug the flower, causing it to burst
7. Go back to the panel across the main room and turn on lights (1,1), (2,2), (2,3), (3,2), (4,4). Visually, that is: ◻◻◻◼ ◻◼◻◻ ◻◼◼◻ ◼◻◻◻

A New Friend (Lily)

1. After being spat out on land, climb the slope and meet Bugsy
2. After the Aidan interlude, grab the vine hanging over the cliff
3. Use Bugsy to climb to the ledge closer to where you started
4. Jump into the flower to go back down
5. Slip past Bugsy, exit the pen, and continue onward
6. After your wings are stolen, continue down the path
7. Pick up the Birrel fur beside the rightmost flower, and take it back to Bugsy
8. Now that Bugsy's pointing to the correct flower (the middle one), enter it

So Much for Lunch (Aidan)

1. At the dam, take the walkway to find the side entrance
2. Climb the ladder to the balcony
3. Use the mask to scare the cat
4. Pick up the key revealed from beneath the flowerpot
5. Use it to open the vent under the window
6. Lock the door using the chain and padlock
7. Open the vent and let the cat in
8. Enter the building through the now open window
9. Pass through the room using the window above the microwave
10. Attempt to cross the ledge, only to be scared off by the bird
11. Take the dropped feather
12. Use the feather on the cat and wait for her to get the bird
13. With the bird out of the way, cross the dam to the lab
14. Enter Nina's lab
15. Leave the lab again to fetch the cavity magnetron
16. Take the lift to…the break room. Oh, dear.
17. Grab the cavity magnetron from the microwave
18. Leave through the window and watch the pie explode
19. Cross the dam again, this time without obstacles
20. Enter Nina's lab again
21. Give the magnetron to Nina
22. Go through the newly opened portal

I found… Grandad… (Lily)

1. Follow the path and learn the truth about Arthur's whereabouts
2. As Aidan, follow the path, led on by "Lily"
3. After focus switches back to Lily and the cutscene ends, go by the ledge and talk to Aidan
4. With Aidan distracting the monster, quickly use Bugsy to climb the ledge
5. Climb the stairs to the balcony where Merr appeared
6. Use the vine to tie a rope to the Illusion Monster
7. Get Bugsy to pick up the rope
8. Jump down onto Aidan
9. Tie the vine to the turret and push the turret off the ledge

His True Intentions (Lily)

1. As Aidan, leave the Illusion Monster behind and head out with Lily
2. After the pair separate, get Lily to enter the shrine
3. Go look at the projection in the pool and watch the fight scene
4. After the cutscene, try to grab the glove from the machine
5. After talking to Nina through the radio, grab the glove again
6. Exit the shrine the way you went in
7. Leave the shrine and head toward the Illusion Monster
8. Use the glove to shoot the gas the Monster is exhaling
9. Try to grab the vine from the Illusion Monster. Roll end!

Time stops when you interact with the Illusion Monster at the end of the game.

I'll update the guide with clarifications or improvements as needed/discovered. Good luck!