So I thought I'd post a basic idea for how aeroplanes could be included in the game in such a way as to fit well both with the aesthetic and the timeline.
First off, a few notes:
1) I'm starting with the beginning point of "What would be a fun addition to the game" and going from there to "How can that be justified inside the game universe" rather than trying to extrapolate the historical development of airplanes based on the discovery of Electroid.
2) I decided to use the Vickers F.B.5 "Gunbus" as the basic model for an early plane. The F.B.5 first flew in 1914, but I would propose that with the advent of Electroid before the turn of the century and its practical application, there may have been MORE military push for the development of aviation weapons. Especially military programs without access to Electroid, yet, may have been scrambling for a means by which to compete with the Leviathan. Considering that the HMS Leviathan flew at the British Exposition of 1908, it seems feasible to me that the development of aeroplanes for war could have been more rapid.
3) I don't think that the presence of conventional aircraft needs to necessarily be exclusive from Electroid based aircraft. Worktroll and others have pointed out that there is not yet developed an electrical source capable of generating the necessary current for Electroid flight on as small a scale as an individual aircraft, but I would assume that the development of such things might be in the wings thanks to folks like Tesla and Edison.
So, the rough draft of the rules, as I envision them.

1) Aeroplanes fly in squadrons which fill one hex. A squadron may consist of several planes, maybe six, but for gameplay purposes, they move as a squadron.
2) Aeroplanes, armed only with something akin to a Lewis gun, do not have armor penetration capability, so they do not roll damage dice against Leviathans.
3) Aeroplanes do not have standard firing arcs, but can only fire in a straight line down the hexrow they are facing.
4) Aeroplanes have a much shorter attack range, reflective of roughly 800 yards (the approximate effective range of the Lewis gun, I couldn't find a direct correspondence between hexes and yards specifically listed. Can anyone clear that up for me? Maybe 3 hexes?)
5) Due to their speed and maneuverability, Aeroplanes move AFTER Leviathans.
6) Attacks by Aeroplanes take place AFTER all Leviathans make their attacks.
7) When resolving attacks by Aeroplanes against Leviathans, a squadron of Aeroplanes roll 3 white location dice. If any location die rolls a location that has already been destroyed, follow standard game rules for Critical Damage (Lieutenant's Manual, p. 21)

When resolving attacks by Leviathans against Aeroplanes, simply roll the location die. In the event of a hit, cross off one of the Squadron's location dice. When all three dice are crossed off, the squadron is destroyed.
9) When resolving attacks by Aeroplane Squadrons against other Aeroplane Squadrons, the attacker recieves DOUBLE his current Location dice. If attacking from directly behind at a range of 1 hex, the attacker recieves TRIPLE his current Location dice.
The culmination of these rules should produce a basic concept of Aeroplanes which have a role in the field of combat, but who are entirely dependant on the Leviathans to be able to damage another Leviathan.
The fact that the Aeroplanes attack AFTER the Leviathans means that the Leviathan has a chance to direct its fire at the attacking squadron and limit its effectiveness BEFORE the Aeroplanes fire.
Doubling and Tripling the Location Dice for Aeroplanes firing on another Squadron of Aeroplanes is meant to simulate the fact that their weapons are more specifically designed to attack small targets that are moving around a fair bit.
As far as progression of technology goes - it might be better to start off the early Bi Planes with only 2 Location Dice for Damage and Integrity, and then grant later planes more location dice, etc. Eventually, technology may grow to the point where aircraft use weapons that do actual damage - but I definitely don't expect so as early in the timeline as Leviathans begins....
I definitely welcome any thoughts, comments, or contributions. This is a Creative COMMONS, after all. Who knows, maybe we can add something fun to the game together?
The image used above (and attached) is copied from the Lieutenant's Manual .PDF and should only be used withing the bounds of the Creative Commons License.