Both abilities are activated abilities, and you can only activate one ability at a time. You can't pay the activation cost for one ability and use that same payment to activate another ability just like you can't use one mana to pay for two different spells.
Basically, Wild Growth works when you tap for mana because it's an ability that triggers when you tap the land for mana. Squirrel Nest and Skygames actually give the land a second activated ability, and you have to choose between that and the original tap for mana ability. It's just like how Abuna Acolyte has two abilities on it that you activate by tapping the Acolyte. You have to pick one, because you can't tap once to do both.