Compatibility works in following fashion. 

All the armors added by Forgotten armory are craftable by combining multiple layer armors, creating single piece occupying all the slots. The main Reason why is visual clipping. The armor models are too small for fitting with Combat Overhaul extensions, middle and skin layer parts would stick out making it look buggy. I tried resizing but personally im just not a big fan of larger models and i ended up not satisfied with the final product and look of scaled "fitting" models.

This is the simplest and smoothest way i know how to implement it now. I plan to make alternative patch, where the armor will occupy only one or two slots and the clipping will be managed manually by the player by using another dependent mod called "Vanity". Also, i made up my own "FA lore" reasons for this, so if anyone is interested in the short stories on the bottom of the page, you can read about "Master Eldred reasoning".

Loud disclaimer. I dont use Combat overhaul in my gameplay, i have no idea if the armor is balanced properly or overpowered. This is going to take time to flash out based on your input. Also i expect some buggery to be happening so let me know. I will fix what i can. 

Tested compatibility with 
Forgotten Armory_Templar v1.2.0
Forgotten Armory_Greenwich v1.2.1
Forgotten Armory_Gothic v1.1.1
combatoverhaul_0.3.9
overhaullib_0.2.10

Hope you enjoy!

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Roadmap

(without any commitment and subject to change)

Greenwich style (plate armor variation) - DONE
Gothic style (plate armor variation) - DONE
Templar style (plate, scale, mail armor variation) - DONE
Combat Overhaul compatibility (Eldred Patch DONE, Thomas Patch WIP)
Landzknecht style (brigandine armor variation)
Hussar style (brigandine, plate armor variation)
Jousting style (plate armor variation)
Swiss guard style (plate armor variation)
French Gendarmes style (plate armor variation)
Multiple helmet variations for every style
Adding weaponry for each style

More Mail, Scale and Brigandine "variations"
... and anything that comes to my mind during development or maybe is suggested by you guys :-) 

I will do my best to push theese regularely but i can dedicate to modding only limited time due to busy life so im not sure how fast i can push theese out.

From the Memoirs of Grandmagister Cardinalis Thomas IX. The Pure

[Scene: The forge. The fire crackles. The apprentice, THOMAS, approaches his master,
the legendary armorer, MASTER ELDRED.]

Thomas:
Master Eldred… may I ask you something?

Master Eldred (without looking up):
You already did. But go on, lad.

Thomas (hesitant):
It's just… I’ve been watching the knights from the eastern marches.
Their armor, master. They wear gambesons, mail, and plate — all layered as they see fit.
They swap pieces depending on the battle. Our suits… they’re always whole. One piece.
Fixed. Isn’t that… limiting? Shouldn’t we give them more flexibility?

[Master Eldred slowly lowers his hammer, turning to Thomas with wide, incredulous eyes.]

Master Eldred:
Limiting!?
By the forge, Thomas, listen to yourself! Flexibility?
You speak as if we’re stitching together patchwork quilts for grandmothers!

Thomas (sheepish):
I didn’t mean to offend, master, but wouldn’t a knight want to choose?
Mix a bit of mail here, some plate there… depending on what they need?

Master Eldred (voice rising):
CHOOSE!?
Do you think a master painter lets the client pick which corner of the canvas should be green?
Do you think a symphony conductor asks the audience which instruments should play the melody?
No, Thomas! Art demands unity! Precision! Balance!

Thomas (softly):
But master… wouldn’t having options be better for them on the field?

[Master Eldred strides across the forge, eyes blazing, arms wide.]

Master Eldred:
The field of battle, Thomas, is no place for options! It’s a place for certainty! For perfection!
Every line of my breastplate — every curve of the vambrace — every gap, every rivet — calculated! Balanced to perfection!
You think you can just slap a gambeson under my plate and not throw off the entire articulation!?
Do you know what a quarter inch of extra padding does to the shoulder rotation!?
Do you!?

Thomas (stammering):
I-I suppose it changes the angle—

Master Eldred (booming):
IT DESTROYS IT!
The pauldron binds, the couter catches, the gauntlet misaligns!
The whole suit sings in harmony, boy. Each piece dances with its neighbor. This is not… mix-and-match haberdashery!
This is the symphony of steel!

[He grabs a gleaming cuirass from the stand, holding it up like a sacred relic.]

Master Eldred (reverently):
This… this is not armor.
This is certainty.
When my knight strides onto the field, his enemy faces the full force of my work. Seamless. Uncompromising.
He does not wonder if his spaulder will slip because some fool decided to wear an extra layer of padded rag beneath!
NO!
When I forge armor, Thomas, I forge truth itself.

Thomas (in awe, but still timid):
But… what if the battle calls for different protection? Against arrows, or axes, or—

Master Eldred (interrupts, furious):
Then I build a different suit, you impudent squirrel!
Tailored to that battle. Designed from the ground up. Not some abominable Frankenstein of random pieces!
Every joint, every weight, every tension point tuned as a master luthier tunes his violin.
You think that tunic-selling street rats know better than I!?

[Master Eldred leans in, intense and wild-eyed.]

Master Eldred (low voice, almost whispering):
Control, Thomas.
In my forge, there is only control.
Mixing layers like a soup pot? That’s for amateurs. For dabblers.
We are architects of war.
And I will not have my symphony corrupted by the chaos of choice!

[A long silence. The forge crackles again.]

Thomas (finally, with a tiny smile):
You’re… quite passionate about this, master.

Master Eldred (suddenly beaming):
Passionate?
Oh no, lad.
I’m right.