Why does it take several seconds of time for the photodiodes to come out of saturation

In hole movement, the particles that are moving are still electrons. When an electron moves to the conduction band (i.e. at any temp above 0 K), there is an empty state that is created in the valence band that was originally occupied by the electron. This empty state is the hole. If another electron from the valence band moves to occupy this hole, it creates another hole one atom over. A chain of such electron movements could be thought of as a hole moving.

I think this is simplifying things quite a bit, but it gets the gist, and I don't really understand the more advanced version. :)

Edit : Actually, I'll try to include the advanced version, by just quoting from the excellent Daniel Mittleman from this awesome thread on the very related topic of 'Are holes real?', since most people probably can't access that thread.

... no such thing as a lone electron or, ... a lone hole - inside any solid. Any charged particle will interact with all of the other electrons, and nuclei, in the solid ...

... [With] These interactions .. taken into account ... One ends up describing what are called 'quasi-particles', which are excitations of the solid that, in some way, resemble a lone electron or a lone hole ...

It is not just semantics - [electrons and holes] are both equally real excitations of the many-body state of the solid.

So, while the simple picture generally enough for getting the drift of things, there's more beyond it, and there's a bit more to holes than just the absence of electrons.

It can be confusing because in a MOSFET the saturation region is something else and they call the "linear" region what would be the "saturation" region in a BJT. Why oh why?

Here's my simplified picture of things for a BJT: -

Note that all the curves for different base currents do not overlap as is commonly shown. If they did overlap there would be no BJT based 4-quadrant multipliers (Gilbert cell). They rely on the saturation region being able to modulate the current for a given CE voltage. Anyway, that's a bit off the mark for your question.

The saturation region does include the scenario when CB is forward biased but I don't think this is particularly helpful - the saturation region (or close to it) must still encompass normal transistor amplification and, as far as I know, this cannot happen when collector and base are forward biased.

Why doesn't further increase in base current cause changes in collector current?

It does up till the point when the collector-base junction is forward biased. The curves look bunched in your diagram (and this is an error basically) but they are still different and for a given low voltage across C-E, the current is proportional to that voltage AND the base current.

Hope this helps.