I don't play archipelago maps often. I get the sense from comments made by top-notch players that the AI suck on archipelago maps. So you may not want to play a low seas, snaky islands. Obviously, plan on building several transport vessels. Since Churchill begins with fishing, this will help if it is the route you choose.
Agreed, use high sea level and small islands map option instead.
Beeline to Sailing first, so you can build Galleys and maybe The Great Lighthouse. Galleys will be your only naval unit for a long time, though Metal Casting at some later point would be good to build a few escorting Triremes. Also, consider building The Colossus after completing Metal Casting, especially if Copper can be connected.
Chariot rush. Requires hunting/agriculture, An Husb, and Wheel. Maybe not the best option, but a two move unit helps with quick speed.
Axe rush. Requires BW on a river or BW and Wheel. This is doable, but one move units are hurt by quick speed.
Horse archers war (clearly not a rush)/ Requires hunting, An Husb, wheel, archery, and Horseback riding. Expect a few spearmen by time you get your horse archers going.
It will be much easier to find Iron then either Copper or Horse (Horse is a via option though), so your early rush units will be Swordsmen with an occasional Axeman or even Spearman. The AI will also find Copper hard to find, so it may have mounted units and archers as defenders, when your early rush begins.
On an Archipelago map with Tiny Islands, mounted units aren't all that useful, since naval units can unload your military units adjacent to cities (blocking reinforcement from a the occasional city that may be on the same island). I would focus on building melee units. Note that with the Tiny Islands option, the vast majority of islands will have at most one City, since there usually isn't room and the AI doesn't try to pack cities close together, making multiple city islands even rarer than need be.
Catapult/Elephants (Also not a rush). This may be the best option for you. Clearly you need Ivory. Possibly try Oracle slingshot for construction.
This can be a great option, since you can start with a map that has Ivory. Although Elephants and Catapults are not an early rush, the units can still be rushed out and Elephants are very effective against Civ that lack Iron Working or connected Iron (again Copper is usually rare enough to ignore).
Maceman/Trebs. If you can Oracle CS and tech to Engineering and Machinery, this may give you the tech lead you need so your units won't obsolete too quickly.
You can use these units for your final rushes, before focusing on your Diplomacy and Mass Media goals (don't forget to generate a Great Engineer to build most of The United Nations).
At this point, you will want Astronomy to build Galleons. Its easier when you keep up with naval technology.
You can always sprint to cannons or cuirassiers, but then you are deviating to tech that are not needed for Mass Media.
I would say that when Macemen and Trebuchets are too obsolete, make peace and focus totally on Diplomacy/TUN.
Sun Tzu Wu