I'm going off the lvl 10 bardic spell feature here, and my bard has bracers of archery and has been killing it when he's been using his bow.
The thing that makes me scratch my head about elemental weapon is that it deals
1d4 damage when the weapon hits. Note that it does not state that the ammunition becomes magical, just that it deals extra damage on a hit.
So would arrows fired from a bow that has elemental weapon cast on it continue to deal 1d4 elemental dmg if my character was to then cast Swift Quiver?
You are right, a magical bow will not make the arrows it fires into magic items. Nowhere in the spell's description or in passages about magic weapons can we find anything like this. But even if it was so, it would not interfere with swift quiver:
You can use a magical bow with nonmagical ammunition without problem. The arrows (or bolts) produced by the spell will work with a magical bow (however it was made magical) and get any bonus granted by the weapon.
Note: Both spells require concentration, so you will need someone else to cast (and maintain) one of the spells.