Time travel is something completely unprecedented within MGS and further pushes the two away. In some weird universe, yes they coexist, but it is clearly a no from here. I will not budge on this. No Kojima written plot? No canon.
I mean, tell me. How does the colorful world of Ape Escape fit in to a world with this as a significant part of it? May I present the War has Changed quote from MGS 4.
War has changed.
It's no longer about nations, ideologies, or ethnicity. It's an endless series of proxy battles, fought by mercenaries and machines.
War--and it's consumption of life--has become a well-oiled machine.
War has changed.
ID-tagged soldiers carry ID-tagged weapons, use ID-tagged gear. Nanomachines inside their bodies enhance and regulate their abilities.
Genetic control, information control, emotion control, battlefield control…everything is monitored and kept under control.
War…has changed.
The age of deterrence has become the age of control, all in the name of averting catastrophe from weapons of mass destruction, and he who controls the battlefield, controls history.
War…has changed.
When the battlefield is under total control, war becomes routine.
I know you want Ape Escape to be some serious spanning thing, but it's not. It's a game some have some nostalgia for but was ultimately unsuccessful, which is why a Playstation Move game was it's last. Then again, Metal Gear's last game was Survive, so maybe that argument isn't too sound.
Besides the point. The two are unconnected, at least the versions presented to us. If Comic would like to say the universe Snake comes from (Given they are in different times) is ultimately connected to Ape Escape, he can do that. Not that he'd meet Specter before his untimely demise.
tl;dr, No. I said it already, the Ape Escape crossovers were jokes. There's more evidence weighed against them then for.