Nice looking root cellar you have built! It looks like the roof is made of wooden timbers that span the concrete walls, I am wondering why you opted not to pour a concrete lid for it? I built a cellar with very much the same design, only a little smaller. Mine is only like 6 feet wide, so with a 4" slab with LOTS of rebar, I piled about 3 feet of dirt on top.
I too have been thinking a lot about cellars as fire shelters. One thing I would like to point out: you should make some provision for what you are going to do if a tree falls on that outer hatch. Electric chainsaw, maybe? In a pinch, you could chop away at your ceiling with an axe, maybe?
Anyway, on the topic of fresh air in confined spaces; I have done some reading. The general consensus seems to be that CO2 concentrations will kill you faster than lack of oxygen. From what I gather, this guy got most of the details about right:
He uses 30,000 ppm as the lethal cut-off, but i found some other sources that said that this is likely too low. It would probably take closer to 50,000 or 5% CO2 by volume to cause incapacitation in about half an hour. In general, his numbers check out, and can be simplified to this: every 10-15 cu ft of air will last you about an hour. Counting your blocks, it looks like you have an 8'x12' space with 8' ceilings - so 768 cu ft. Using a fairly conservative 15 cu ft/hr, one adult should be able to survive sealed up in there for over 50 hours. 2 people could easily go a whole day. This volume will decrease as you store stuff in there, so dont forget to take that into account. I have never experienced a forest fire firsthand, but I understand that they burn through an area pretty fast. I suspect you should have no problem at all sheltering in there for several hours. I think if you could seal off the air inlets from the inside you should probably be fine.
Another thing worth mentioning is about fires "consuming all the oxygen." From what I have read, fires in solid materials need a certain oxygen content to even burn. If the air becomes too depleted of oxygen, the fire will go out. This might be different for gaseous fuels, so dont store any gasoline in your shelter! It sounds to me like a human can survive on less oxygen than a wildfire can, but having a way to seal yourself off from the fire seems like the best bet. There is also going to be an insane amount of smoke, so really get after all the cracks with some duct tape. As for the scuba tanks, they are going to buy you a few hours, at best, right? A more compact method would be to store ordinary welding oxygen. An 80 cu ft tank costs like 20 bucks to fill, and should be enough for 1 person for 4+ days. You would then need to also scrub CO2 out of your air, but CO2 scrubbers can be put together out of Ca(OH)2 with a trace of NaOH on it. You can buy 50lb bags of slaked lime at home depot for 12 bucks. I forget how much CO2 that would remove, but it was a lot. Youd need a fan, and some sort of container to hold the little clumps of co2 adsorbent. Youd also need to be a little careful here about raising the O2 content too high. I would not suggest you try that without a dependable O2 meter handy.
Anyhow, I am chipping away at a new cellar project. I will have to put together a post about it when it gets a little farther along.