No, assuming you have legally activated the Talisman, your opponents do not have time to stop you from drawing a card from the Well's ability by destroying the Well.
The rules define what a mana ability is (emphasis mine):
605.1. Some activated abilities and some triggered abilities are mana abilities, which are subject to special rules. Only abilities that meet either of the following two sets of criteria are mana abilities, regardless of what other effects they may generate or what timing restrictions (such as “Activate this ability only any time you could cast an instant”) they may have.
605.1a An activated ability is a mana ability if it meets all of the following criteria: it doesn’t have a target, it could add mana to a player’s mana pool when it resolves, and it’s not a loyalty ability. (See rule 606, “Loyalty Abilities.”)
So the fact that you gain life doesn't change the fact that the ability is a mana ability. So it doesn't use the stack. So once you activate it, the next time anyone gains priority, you have the mana, you have gained the life, and the Well has triggered and that triggered ability has been put on the stack, as detailed in
605.3b An activated mana ability doesn’t go on the stack, so it can’t be targeted, countered, or otherwise responded to. Rather, it resolves immediately after it is activated. (See rule 405.6c.)
405.6c Mana abilities resolve immediately. If a mana ability both produces mana and has another effect, the mana is produced and the other effect happens immediately. If a player had priority before a mana ability was activated, that player gets priority after it resolves. (See rule 605, “Mana Abilities.”)