While enchanted, the next successfully used skill will end the enchantment. If that skill was a spell, Arcane Echo gets replaced by that spell for 20s. The "new" spell is ready to use with its normal costs, activation, and recharge.
"Enchantments last longer" only makes the Arcane Echo enchantment longer, not the copied spell.
When the copied spell reverts to Arcane Echo after 20 seconds, Arcane Echo begins its 20 second recharge.
This spell interacts favorably with Auspicious Incantation, because the extra recharge time caused by that spell will not affect copied skills, nor will it affect Arcane Echo if you copy a spell with it.
For copied spells that can only be cast once in their 20 second lifespan because of long recharges, use Glyph of Sacrifice to cast them instantly. The added recharge time won't make a difference since the spell will revert to Arcane Echo and not affect its recharge.
There seems to be a bug when commanding heroes to use the copied skill. If the skill reverts back to Arcane Echo before the hero can use it, the AI will stop using any skills at all while it tries to use the now non-existant skill. To remove this, order the hero to use another skill so the checkmark appears on a different skill than Arcane Echo and then cancel that skill by clicking it again. The hero will resume normal activity.