I beg to differ.  To start off, a good Gifts player doesn't screw up but rather wins.  Second, part of being a good Gifts player is knowing your deck perfectly and knowing before you even cast Gifts what cards you're going to get and what result they will have.  The way I solved the puzzle is by simply laying out the identical situation with my own Gifts deck at which point the solution just popped into my head.  Tinker/Time Walk is in fact a much more dangerous play, because it gives the opponent an extra turn.  If by some chance they bounce the Colossus then you've wasted not only a turn but one of the most potent spells in the deck (namely Tinker).  The easy play is not always the correct play.

Notice how the original question posed by Steve was whether or not it would be possible to win this turn.  If you don't give your opponent a chance to live, they can't give you a chance to die.  Plus, given the opponent's board, it's a fairly safe assumption that if you can win, you will (ie no counterspells).

If I'm still missing something then explain this question to me: Why win in a few turns when you can win NOW?

Here is what you do:

Put back Force of Will and Mana Drain.

Tap Sol Ring and Mana Vault, play Tolarian Academy and tap that as well.    floating.

Tinker, sacrificing Mana Vault, for Black Lotus and sac lotus for blue.    floating.

Merchant Scroll for Gifts Ungiven and cast it.  floating.

Gifts for:
Lotus Petal
Mox Ruby
Mox Jet
Yawgmoth's Will

There are two basic scenarios.  Cast one of the mana sources they give you and play Time Walk. 

It doesn't matter what they give you, just don't sac Lotus Petal.  The only time you'd sac Lotus Petal is if it were the only mana source, meaning they gave you Yawgmoth's Will in hand, in which case it wouldn't matter.  Also, they are forced to give you either Yawgmoth's Will or a red source, and you already have the black source with the Underground Sea in hand.

For the sake of discussion, we'll assume the scenario that makes it most difficult to combo out.  They give you Mox Ruby and Mox Jet.

You move to your time walk turn.

Play Mox Ruby or Mox Jet (whichever you didn't cast last turn when you played Time Walk), and Underground Sea.  Spell count 1, nothing floating.

Tap Sol Ring, Mox Ruby, Mox Jet, and Island to cast Recoup targeting Yawgmoth's Will and Yawgmoth's Will.  Spell count 3, nothing floating.

Cast Black Lotus, Lotus Petal, and Mana Vault off the Underground Sea.  Spell count 6, nothing floating.

Tap Tolarian Academy and Mana Vault.  Spell count 6, floating.

Merchant Scroll for Mystical Tutor, cast Mystical Tutor getting either Burning Wish or Tendrils if you run it.  Spell count 8, floating.

Brainstorm, getting Tendrils/Burning Wish in hand.  Spell count lethal (will be at least 10 after Tendrils is cast),   floating.

Sac Lotus for Black and Lotus Petal for Red.  Spell count lethal, floating.

With that, you have enough mana of the appropriate color to cast either Tendrils of Agony or Burning Wish followed by Tendrils of Agony.

GG.

edit: Steve, we need to talk about my Gifts list.  I think you will like it, and frankly, I am an EXTREMELY profecient gifts player, as I was playing 4 Gifts TPS before MDG even existed.

You're comparing apples to oranges.  If you're using Memory Jar, you're going for the Tendrils kill and Colossus isn't even an issue.  The point of using memory jar is specifically that you do NOT have to use Yawgmoth's Will, hence the comment that it strengthens the deck against graveyard hate. 

Then the play is no good.  Using up all your resources and then crapping out != tech.

Granted my list of MD gifts is slightly different than yours, here is what I would do:

Put back Recoup and Rebuild.  Why?  Recoup won't be necessicary and I won't need rebuild for spell count.  The Volcanic is more important because it'll give you one extra blue mana when you're going off which is usually more important than excess colorless.

Next, you play Mox Ruby, Merchant Scroll for Mystical Tutor, and pass the turn.

On your next turn, you use the Mox Sapphire to Mystical Tutor on your upkeep for Yawgmoth's Will.

You play the Volcanic Island, then the Mana Crypt into the Mana Vault, then tap the Mana Vault.  4C floating.  Tap the Island to play Gifts Ungiven: C floating, 4 spells.

You get:
Black Lotus
Dark Ritual
Mox Jet
Lotus Petal

Now I must pause for a minute.  I run 1 Dark Rituall main board because I feel it immensely strengthens my gifts piles.  If you don't run one like I do, you would've put back the Volcanic Island and kept the Rebuild off the Brainstorm, and instead of getting the Dark Ritual with Gifts you would've gotten an Underground Sea.  Either way, you win this turn.

The smart player obviously gives you Mox Jet and Lotus Petal, because either the Dark Ritual or the Lotus Petal will give you 2 spells when played through Yawgmoth's Will, the Lotus Petal simply gives you less mana.  You proceed to play Mox Jet and Lotus Petal.  Tap the Jet and Ruby and use the Petal to play Yawgmoth's Will.

Spell count 7, Volcanic Island still untapped.

Petal into Dark Ritual, Lotus for U, spell count is now lethal.  You have floating.  Use to Mystical Tutor for Tendrils, Brainstorm into the Tendrils, and use the rest of your mana floating to win.

Now there are a few things to consider.

1. Most MD Gifts lists do not run Dark Ritual.  As I explained above, this is fairly irrelevant to this scenario.

2. Question: What if they have a counterspell?  Answer: Than they'll counter a spell no matter what you play.  Sure you could play on the safe side and Merchant Scroll for Ancestral Recall, but they could just counter the recall.  Against fish, you'll be fighting an uphill battle either way, so you may as well race.

3. Many MD Gifts lists do not run Tendrils main.  I feel this is a huge mistake.  Running burning wish is clunky.  It makes it more difficult to go off because you require an extra .  This can create a huge problem because if you're trying to go off turn two, you may only have one land drop.  If that land is a fetchland, you want to be able to fetch for an Underground Sea and not worry about losing because you're short the . 

On a side note about win conditions, Darksteel Colossus is also a problem, because it is another unnecessary dead card.  The Tendrils kill, when wielded by a competent player is just as effective as Darksteel Colossus, because they both result in a dead opponent.  The difference is that running Tendrils main speeds up your kill by a turn or two and you don't have to worry about drawing Colossus.  Then you can either cut Tinker and Colossus entirely or run Memory Jar in place of the Colossus to strengthen your deck against graveyard hate.

Well, you'd need gemstone cavern, card to rfg, and quicken plus two more cards for pandeburst and another for eureka.  Is there one card that plays for free, as an instant, and gives you four green?  edit: you get two cards, one as the draw from quicken

Creatures:
4 Wild Mongerel
3 Aquamoeba
4 Basking Rootwalla
4 Wearbear
4 Ninja of the Deep Hours
4 Soldevi Sage

Spells:
4 Accumulated Knowledge
4 AEther Burst
4 Tolarian Winds

Uncommons:
3 Wonder
2 Aether Vial

edit:
10 Island
10 Forest

This deck is extremely aggressive and loves playing creatures at the end of the opponent's turn.  Evasion is everything in peasant and the deck has a very obvious mana curve, so drawing any of the uncommons is house.  The deck has an incredible amount of synergy and builds exponentially especially once Soldevi Sage gets online.  The AEther bursts are more for the early game because late game the deck is simply too aggressive for most decks to handle.  The mana base is also great because any hand with at least 1 Forest and 1 Island is keepable.

There are virutually no peasant tournaments but in testing it consistently beats Affinity and Madness (basically the metagame for peasant) and also has fared extremely well against numerous aggro builds.  The only deck it tends to have a bit of trouble with is burn.  I currently have no sideboard because the deck really doesn't need to sideboard for most of its matchups.  If I had a sideboard it would look something like BEBs, Hydroblasts, artifact hate, and maybe Shelding Plax.

I've also tested the deck with Careful Study, Simic Signet, Coiling Oracle, Deep Analysis, Ideas Unbound, and Obsessive Search and found them all inferior.