Before I lay out the.. "what if, but, then". Let me set a foundation for tanks vs cannons.
Tanks kill cannons in 3 shots. 5 shots for sunkens.
There is a few problems such as...
Not scouting a double/tripple.
A quick reaver drop.
Some sort of devious cheese/strat that will stop the tank push or damage the terrans economy. Crippling the push, or forcing the terran to ALL-IN from there.
~This is still a work in progress~
However. Let's say a protoss chokes + double/tripple nexus.
A single factory tank push would allow for more firebats/stim/range. But... The downside if the protoss goes dragoon range and keeps using 3-6 dragoons to pick off stray tanks.
A double factory tank push requires 2 gas. (2 SCV's on each gas). (3 SCV's on each gas for academy upgrades + firebats).
Theoretically, you can break down the protoss choke, much quicker than a Zerg choke.
This is my approach to tank pushing a choking Protoss. I am up for suggestions. Heck, even a few friendly 1v1's To see what works better, or maybe a tank push vs protoss is completely stupid. (I don't know).