No, the talismans will not work.
It's stated at the end of the story that all magic in the world disappeared along with the Trees.
As for why this happens, it's because of how magic works (described in Chapter 3 of the manga, pages 17-18). When a mage uses magic, their powers are being borrowed directly from a Tree. None of the mages ever have their own magical power.
Basically, talismans are conduits, not batteries. Any magical power that's being stored would still be owned by the Tree of Genesis until it's actually used, with the talisman acting as a conduit between the Tree of Genesis and the person whose skin is touching the talisman.
To give a relatable example: It'd be like if you (the mage) went out and got a refillable Visa card (the talisman) and then put some money (magical power) on it in case of an emergency. The Visa card is "storing" your money, and anyone can use it as long as you're there to sign for it (the mage's voice activation). However, if Visa (the Tree of Genesis) suddenly ceases to exist one day, your Visa card is completely worthless, and all the money it was storing will be gone forever.
This was alluded to near the beginning of the manga, when Hakaze had to ask the Tree of Genesis for power in order to use the talisman to find Aika's killer (chapter 2, page 21 in the manga).
Later (chapter 3, page 18), Hakaze explains that talismans still require voice activation, because the Tree of Genesis won't supply its magical power unless the mage asks for it verbally. It's explained that the main characters are able to use the worn talismans without Hakaze's voice because Hakaze verbally asked the Tree of Genesis for that magical power ahead of time, and then asked for the given power to be activated when the wearer of the talisman invokes it mentally.
So unfortunately for them, the characters of Zetsuen no Tempest won't have the ability to use magic anymore, with or without talismans.
