You would in fact be able to activate Wellwisher's ability successfully on either player's turn. The Wellwisher would be untapped during your untap step; no player obtains priority during the Untap step. The next opportunity to activate abilities is during your Upkeep step. Because the active player has first priority, you can activate the Wellwisher then, paying its cost (tapping Wellwisher); your opponent cannot respond until after this has already occurred, while Wellwisher's ability is on the stack. At this point, Wellwisher's ability (on the stack) is essentially a separate entity, and the Trapper tapping Wellwisher has no effect upon the ability's resolution.
On your opponent's turn, this is no different. Seedborn Muse would untap your Wellwisher as expected; however, your opponent would receive first priority during the Upkeep step, as it is their turn. If they activate Ballynock Trapper's ability targeting your Wellwisher, then both players must gain and pass priority before the ability resolves. When this occurs, you can activate your Wellwisher's ability, which as a cost will tap itself and place the ability on the stack. The Wellwisher's ability will (unless otherwise prevented, by Azorious Guildmage, etc.) resolve, and then the Trapper's ability will resolve, tapping the Wellwisher to no effect.
In either case, at some point, before anything can resolve which would destroy, tap, or otherwise affect your Wellwisher, you can activate the ability; after this, the ability on the stack is effectively independent of Wellwisher, and will not be prevented if Wellwisher is tapped or even destroyed.
There are some exceptions to the previously stated rule -- for example, if it's your opponent's turn a Sudden Death targeting Wellwisher during their Upkeep would prevent Wellwisher from being able to activate its ability while Sudden Death is on the stack, and Wellwisher would be destroyed when Sudden Death resolved. On your turn, however, you could immediately activate Wellwisher's ability (as you have priority); after you pass priority, your opponent could respond with Sudden Death, which would almost certainly successfully kill Wellwisher, but the Wellwisher's ability would still be on the stack and would resolve as expected.
(My apologies for including this as a separate answer; if I could currently comment, my two cents would take the form of constructive criticism addressed at the excellent, but not yet perfected, answer by murgatroid99.)