Well, calculating the exact probability is WAAAYYY too much of a pain, but we can get a reasonably close estimate by ignoring the majority of edge cases.

Hmm... I'll assume that the wiki is correct for a total of 499 spells...

The only way I can think of to generate Mechathun is mindgames, so that's 1/499.

To draw the rest of the deck, there are 30 cards that will definitely draw out the rest of the deck, 2 with exactly 50% chance (wrath/nourish) and 4 more that will do so if the right targets are selected (e.g. slam/mortal coil hitting valid targets or Starfire/UI/Purify not hitting Yogg, Holy Wrath not hitting Yogg or Mechathun) and then branching paths. Altogether, that adds up to about 36/499 chance to empty the deck. Note that if it's UI, then it has to occur AFTER Mindgames (or the board will be filled, so let's just pretend that we can't have that spell get casted). That leaves us with 35/499.

Finally, you need to get Cataclysm, for 1/499. So the chances of getting these three conditions is 2*(1/499)(35/499)(1/499). However, those 3 spells can be any 3 of the 19 spells Yogg will cast.

Finally, we can go an eliminate spells that would automatically kill yogg or Mechathun. There are 8 board wipes (DOOM!, Brawl, Twisting Nether, Poison Seeds, Cataclysm, Enter the Coliseum, Storm Bringer, Evolve). For the sake of avoiding edge cases, let's just remove Cataclysm from the list. There are 9 spells that do 5-8 damage (so can't target Yogg) and 23 spells that do 10+ damage or destroy/silence/transform minions (so can't target Yogg or Mechathun). Together this totals to about 14/499 chance to fail (or 480/499 to not interrupt Yogg).