Excellent sprite work, good turrets, interesting balance. These turrets seem to have far longer range, and less damage than Rampant Arsenal or Bob's turrets. A word of note, it looks like the modular turret damage upgrade is only affecting the heavy machine gun and heavy cannon turrets, when it says on the tech that it affects all of them.

It does affect them all, but HMG and Heavy Cannon have passive +% damage bonus like vanilla Gun Turrets used to have. In the tooltip, it will be listed as "Damage Bonus". Once you research +turret damage for them, tooltip will start listing researched bonus instead, with added "x1" or "x2" to notify turret has passive damage boost as well. If you want to make sure, write down numbers under "Damage" entry, research 1 level of turret damage and compare again, it will go up.

Heavy MG Turret: https://i.imgur.com/hkMcS8k.png has the x2
Gatling Turret: https://i.imgur.com/2PAZpw3.png has no boost
Light Cannon: https://i.imgur.com/aEw56El.png no boost
Heavy Cannon: https://i.imgur.com/LYPr9ga.png x1 boost.

I figure four screenshots is enough to prove what's happening on my end.

That line ("Damage Bonus") only appears if turret has passive damage boost by itself, and confusingly shows researched turret damage bonus if there is one. Instead of looking at that, place down these turrets, hover over them with mouse and look at values in "Damage" tooltip (turret must have ammo). To check if +turret damage applies, simply calculate values yourself.

There are two values, base damage and bonus damage from research, looking like this: "Damage: X + Y <DMG Type>".
Here's the formula:
X = AM * (TB + 1), AM - ammo base damage value, TB - turret passive damage bonus
Y = X * (ADR + 1)* (TDR + 1) - X, ADR - ammo damage bonus (reserach), TDR - turret damage bonus (research)

TB values:
HMG: +50%
Heavy Cannon: +20%
Scattergun: +50%
Others and Vanilla turrets: +0%

Let's use HMG turret as an example. In my game, ADR is +320%, TDR is +300%, AM is 24 (uranium ammo)
X => 24 * (0.5 + 1) = 36
Y => 36 * (3.2 + 1) * (3 + 1) - 36 = 36 * 4.2 * 4 - 36 = 568.8
"Damage: 36 + 568.8 Physical"

Another example with vanilla Gun Turret, uranium ammo, ADR +320%, TDR +470%:
X => 24 * (0 + 1) = 24
Y => 24 * (3.2 + 1) * (4.7 + 1) - 24 = 24 * 4.2 * 5.7 - 24 = 550.76
"Damage: 24 + 550.76 Physical"