I'd recommend giving them iron or steel helmets if you have the metal. The reason is that a metal helmet weighs too little to cause noticeable slow down but will cause a massive boost to your dwarves' survivability in literally every situation except fluid-related incidents (as in burning in magma or drowning in water). Even in a food or drink crisis it'll prevent tantruming dwarves from braining each other. After many forts where I equip everyone with armour I have to say that the metal helms had a far far greater effect than a full suit of bone leather for everyone, and were far less hassle to produce. If you're really trying to get rid of all the ridiculous amounts of metal that you have, I'd recommend mail shirts. Mail shirts+helms still don't cause any noticeable slowdown in untrained civilian dwarves and the mail shirt will prevent edged damage to the dwarves' torsos, upper legs, upper arms, and necks. Which, coupled with a helm protecting the head (from both blunt and edged damage), means that the only parts of the dwarf which are not covered by metal are their lower legs, lower arms, feet, and hands. To pad up this civilian gear I'd recommend bone gauntlets and greaves and leather high boots, as well as leather dresses or robes, trousers, hoods, and mittens.
Also, Crossbows are a better pick for civilian dwarves, as untrained axedwarves even in large numbers will always (>90% of the time) get ripped apart in close combat by a competent opponent but untrained marksdwarves can, if in large enough numbers (i.e. your whole civilian population), severely incapacitate foes before they get into melee range, without suffering casualties. Plus, crossbows are cheaper to make than battleaxes, and are best when made out of cheap old copper rather than your finest steel (higher weight for use as blunt impact weapon). Don't assign squads more than 250 bolts of ammunition, as max squad size is 10 and max quiver capacity is 25, so assigning more bolts just prevents other squads from using them even though the current squad can't either.
If you have surplus weapons I'd recommend giving dwarves "individual choice, melee" as well, on top of crossbows in the uniform list. This will make them drill with their melee weapon when you send them for combat training and have it equipped in their primary hand with the crossbow as their secondary. They will still fire the crossbow, but this way they'll actually use the melee weapon as well. Using individual choice, melee rather than a specific melee weapon gives a greater likelihood of a good mix of damage types among your civilians, which means they won't all be entirely useless against any foe. Battle axes are good all-rounders (and the most likely to be randomly selected by dwarves due to their high price tag) but are outperformed by the other weapons in certain situations, so it is good to have the mix.
And you can set priority of uniform assignments in the military screen, in the (e)quip screen. This lets you re-order squads for the purposes of uniforms, first assignments->last assignments, but you may still want to create empty squads as placeholders (no need to put any dwarves in them though!). The default uniform priorities are based off the order in which the SQUADS were created. The main reason you may want to create placeholders though is that there's no way to change the order of the squads in the (s)quad menu (where you give kill and move orders), so it's usually nice to have your dedicated military at the top of the list.
Oh and last thing: if you want to put miners, hunters, or woodcutters in a squad for admin purposes, give that squad no uniform, and in the (s)chedule military screen make it so that they don't wear a uniform when off duty. This way there will never be clashes, even when you give them squad orders, and they will still do individual drills with their profession's weapon.
Just my two cents from experience with all military forts. Hope that helps