CDR is still an important stat so I don't see Nashor's as a particularly bad item, it's just unsafe early but ironically is the best option available. Protobelt honestly is a very effective first item, but Lich Bane should be replacing Rabadon's directly for AP divers as a third item. The similarity between them and mages makes it so that they both have to be tuned differently despite sharing the items. Spellbinder's currently shares this identity however so Idk what the plan should really be.
We need Lich Bane to have easier AP access but still push it more as a third item. More AP and less mana behind it as a start maybe? Definitely need an AP version of Sheen as part of the build path since base AD and the relevant champs who use Tri Force and IBG actually scale better.
Divers don't really prefer the attack speed from Nashor's as much as the added dueling power. I would rather a new item be introduced that is closer to an Edge of Night for AP divers where you get a channeled shield for a few seconds and primarily is built off of health and AP, but also gives much needed CDR that AP divers miss out on by not taking Nashor's. It should also be more for escapes or approaches rather than fighting.
Hextech Revolver is really good for what it does early on but you can't have that + Nashor's and realistically keep it for the mid game when you need tankier options. Either your Zhonya or Void Staff has to be paid up for in opportunity cost. And most of these champions don't have dual scaling so you can't go Gunblade despite it being ultimately the best option for divers on item active and emphasis on burst.
I'd rather not alienate Lich Bane by making it experience major shifts since it is an additional AP ratio to your best combos. It would be more effective for divers if it recovered mana after the proc or maybe even after takendowns/assists. The problem is that getting it too early is meaningless because it doesn't scale properly like base AD normal Sheen (thus my AP Sheen suggestion) but getting it later just makes it a needless "win more" condition that won't nearly scale as well as Void Staff or Rabadon. If it was a damage amp that you could only achieve after casting a spell and auto'ing thereafter, it would make more sense.
We should probably rethink spellweaving in the sense of cast > AA (rinse+repeat) and instead apply more as it is actually seen for AP divers; casting with occasional autos that undoubtedly still is mostly casting. The even damage spread has to be activated by the auto and maintained by the idea of how quick casts are happening for divers, as they do not auto reset as of current for their best burst scenarios and involve that instead when dueling, which is not going to happen in teamfights. Making an actual storage of bonus damage to involve fast multi-spell combos born from the auto is more realistic than a single-hit interaction, and vastly more approachable for AP divers compared to ranged mages who do not want to be forced to auto's for their best rotations. This also makes sense for when Lich Bane currently is effective and ineffective, though it is obviously still lacking. Other items need to be made to better accommodate this playstyle in general, not just a Lich Bane rework. It's why AP jungling even if you make it out of early game will not largely see much success unless you can work with the current mage itemization.
My last observation on the matter is that mage-like divers only need to be so tanky because they're putting themselves at high risk and cannot escape or keep bursting like Katarina. In fact, Katarina, Leblanc, and Ahri all fit what mage divers should look like; they should have a burst combo as well as an escape to ensure that they won't die for performing to the limit. Even Fizz has this limitation to some extent but also this advantage, so he suffers much less than something like Diana. Hell even Kennen in comparison has enough AoE burst + CC to make his best play have WAY less risk, especially with Zhonya's involved.
Essentially:
-Mage divers need more items, especially ones that add more safety and mobility to the all-in's conclusion since if the kit is designed well enough, it will be able to make a play in the first place without said items.
-Damage spread has to make more sense in terms of how AP divers make their combos and interactions. Higher downtime in exchange for higher play potential. Items must also represent this in terms of what they build.
-AP Diver first item must be equivalent or better than Protobelt for gap closing and doing tons of damage. It is already a good item, though one can debate as to how effective it actually is.
-If possible AP Divers should want to access Presence of Mind to better assist with mana issues.