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This seems good.

I think as long as you're trying to cover CPU and PG in separate skills, you're going to have that second problem if you try to cover that in the skill name.

Weapons Optimization and Advanced Weapons Optimization seems sufficient for both - you're optimizing your weapons' fitting requirements, and opening access to modules that further optimize those weapons.

It's really hard to find something, because the bonus affects one set of items, while the skill unlocks ANOTHER set of items. The best thing I could come up with is:

Weapon Modules CPU Management

Kinda implies it boosts the CPU management of weapons, and at the same time mentions weapons modules... It's ambiguous, but you're not going to find something not ambiguous in four words or less when you have at least two conditions within your skill definition.

Advanced Weapons Powergrid Management

... Yeah, I think you should actually stick with the old names :p
Or change it to "Weapon CPU Management" and "Weapon Powergrid Management", disregard the fact that it doesn't adress the weapon modules prerequisite. After all "Weapon Upgrades" doesn't adress the CPU part of the skill.

Edit: If you really wanted a good name, I'd suggest "Damage Upgrades & Weapon CPU Management". But that'd be a first in terms of length and in that it has a "&".

The skill pre-req changes that Fozzy has listed (so-far) are good changes.
The reasons to make the changes are sound in that it will make the game more consistent.

As for Naming? Trying to describe both sides of the coin in the name seems to produce inelegant\clunky solutions.
The current names seems to be the best option right now.

How about "Computerized Weapon Upgrades"?

Does more to indicate the CPU boosting nature of the skill without losing the indication that it also relates to modules. Also potentially adds a nice bit of flavor about how weapon upgrades work.

What's GMP anyway?

AWU and EWAR skill changes are needed. The other two - not really.
Please leave the name for WU as it is. It's described in many tutorials that new players will read. Unnecessary confusion is not what they want.

As I told you on twitter,

Why not split the skill, the problem currently is that newer players need to train awu for fitting purposes.

Since WU only helps with the CPU side of things and AWU with the Power Grid, why not split the skills up?

Have two skill tiers, one for CPU and one for Power Grid

Lower the basic multiplier and then further the skill in a specialization one, maybe even allow for an even better end result?

Do be careful with the last suggestion though

I like these two names personally.
Weapon Calibration (CPU)
Weapon Installation (PG)

Weapon Support Systems Optimization

Weapon Powergrid Efficiency

Weapons upgrades.
Weapons optimisation.

Definitely yep to the second tier of e-war and GMP being easier to get into. They are pretty important. Not bothered about nanite stuffs.

Weapon Upgrades --> Weapon Resource Usage Optimization

Combine the reduced CPU and PG usage from the current skills, reduces CPU/PG usage by current percentage per level. Keeps current WU skill prereqs.

Advanced Weapon Upgrades --> Weapon Upgrades (a little clunky renaming to a name currently used, I know)

Grants access to weapon upgrades, increases effectiveness of weapons by current WU percentage per level, level 4 required for T2 weapon upgrade modules. Skill prereq: WRUO 1-3?

Being that skill effects are being split/moved, you'd probably end up having to grandfather in current skill levels by giving away skill points in some cases. Probably not a huge deal given the current trend of decreasing difficulty in acquiring SP anyway ¯\_(ツ)_/¯

For WRUO, grant the higher of current WU or AWU (i.e., if someone currently has WU 5 and AWU 4, they'd essentially be getting the equivalent of AWU 5 for free).

For WU (after rename), they would get their current level in WU (prior to rename).